POLYPHASE  CURRENTS 


TILL 


POLYPHASE    CURRENTS 


POLYPHASE 
CURRE  NTS 


BY 

ALFRED    STILL 

ASSOC.M.INST.C.E.,   M.I.E.E?,  MEMBER  A.I.E.E. 

AUTHOR    OF 

"ALTERNATING  CURRENTS  AND  THE  THEORY  OF  TRANSFORMERS"  AND 
"OVERHEAD  ELECTRIC  POWER  TRANSMISSION  " 


SECOND  EDITION,  REVISED 

WITH  101   ILLUSTRATIONS 


THE  MACMILLAN  CO. 

64   AND   66    FIFTH    AVENUE,    NEW   YORK 

WHITTAKER  &  CO.,  LONDON 

1914 


v\\U 

O  o 


PREFACE  TO  SECOND  EDITION 

IN  revising  this  volume  for  a  new  edition,  the  scope  and 
view-point  have  not  been  changed.  It  is  not  claimed 
that  the  treatment  of  the  subject  is  exhaustive;  but 
an  attempt  has  been  made  to  present  the  principles 
underlying  the  operation  of  polyphase  currents  in  clear 
and  simple  terms.  The  student  whose  aim  it  is  to 
become  proficient  in  the  design  of  polyphase  machinery 
may  use  the  book  only  as  an  introduction  to  more 
advanced  works ;  and  those  requiring  practical  informa- 
tion on  such  details  as  switchboard  connections,  or 
descriptions  of  actual  machines  and  apparatus,  must 
seek  this  elsewhere ;  but  nevertheless  the  needs  of  the 
practical  engineer  have  been  constantly  in  the  writer's 
mind. 

The  addition  of  new  matter  has  been  deliberately 
avoided  so  far  as  possible  ;  but  much  of  the  original 
matter  has  been  rearranged,  rewritten,  or  entirely 
omitted.  The  assumption  that  the  reader  has  a  fair 
knowledge  of  continuous  currents,  but  is  unfamiliar 
with  alternating  currents,  is  still  made,  although  there 
is  perhaps  less  reason  for  it  now  than  at  a  time  when 


VI  PREFACE    TO   SECOND   EDITION 

comparatively  young  engineers  had  left  the  colleges  or 
universities  before  the  subject  of  alternating  currents 
was  adequately  taught.  The  writer  believes,  however, 
that  the  omission  of  the  two  introductory  chapters  would 
detract  from  the  value  of  the  book. 

In  the  matter  of  minor  alterations,  it  may  be  men- 
tioned that  almost  every  figure  or  diagram  has  been 
redrawn ;  all  vector  diagrams  have  been  reversed  to 
accord  with  the  internationally  agreed  convention  re- 
garding leading  and  lagging  vectors ;  and,  for  the  same 
reason,  the  symbols  used  to  denote  certain  physical 
quantities  have  been  altered. 

PURDUE  UNIVERSITY, 
LAFAYETTE, 

INDIANA,  U.S.A. 
1914. 


PREFACE  TO  FIRST  EDITION 

WITH  the  extended  use  of  polyphase  alternating  currents 
for  the  transmission  and  distribution  of  electric  power, 
there  would  seem  to  be  a  demand  for  a  book  treating  of 
the  theoretical  considerations  involved  in  polyphase 
working  in  such  a  manner  as  to  commend  itself  to 
practical  engineers  and  those  students  who  are  without 
the  mathematical  knowledge  required  for  the  study  of 
the  more  advanced  works  on  the  subject. 

The  author  has  adopted  a  non- mathematical  treatment 
of  the  subject  throughout,  but  has  made  extensive  use 
of  graphical  methods.  The  introductory  chapters  are 
written  with  the  object  of  explaining  the  use  of  vectors 
in  solving  alternating- current  problems,  and  also  for 
the  purpose  of  drawing  attention  to  the  chief  points 
of  difference  between  alternating  and  continuous  currents 
of  electricity.  It  should,  therefore,  be  possible  for  any 
student  or  engineer  with  a  fair  knowledge  of  continuous- 
current  working,  but  with  only  a  superficial  acquaintance 
with  alternating  currents,  to  obtain  a  thorough  ground- 
ing in  the  principles  underlying  polyphase  working, 
provided  he  will  take  the  trouble  to  master  the  contents 
of  the  first  two  chapters. 

The  few  notes  added  at  the  end  of  the  book,  in  the 
form  of  an  appendix,  deal  with  certain  matters  referred 
to  in  the  text,  but  which  are  not  essential  to  the 


vill  PREFACE   TO   FIRST   EDITION 

scheme  of  the  book,  or  to  the  elucidation  of  subsequent 
chapters. 

It  may  be  urged  that  the  subject  of  electric  power 
transmission  has  been  given  undue  prominence ;  but 
the  reason  for  the  somewhat  disproportionate  length  of 
this  section  is  that  the  book  has  not  been  primarily 
written  for  the  use  of  designers  of  polyphase  machinery 
(for  these  must  have  a  very  thorough  and  detailed  know- 
ledge of  the  subject),  but  rather  for  the  user,  or  the 
engineer  who  has  to  lay  out  and  to  work  complete 
polyphase  schemes.  If  exact  details  are  required  of  any 
particular  piece  of  polyphase  machinery,  this  information 
can  always  be  obtained  from  the  makers  of  the  plant. 

Again,  Kelvin's  law  and  the  principles  determining 
the  most  economical  section  of  conductors  are  discussed 
at  some  length,  not  only  because  they  have  a  very 
important  bearing  on  the  subject  of  polyphase  trans- 
mission of  power,  but  also  because  they  do  not  appear 
to  be  generally  understood.  These  questions  have  been 
treated  in  a  non-mathematical  manner  ;  but  the  main 
points  at  issue  have  been  clearly  stated,  and  the 
reader  should  have  no  difficulty  in  calculating  the  most 
economical  size  of  conductor  to  suit  any  given  electric 
transmission  scheme. 

The  author  desires  to  thank  the  editors  of  the  Electrical 
Engineer  and  of  the  Electrical  Review  for  the  loan  of  blocks 
and  permission  to  reprint  portions  of  certain  articles  that 
have  appeared  in  these  journals. 


ELLESMERE  PARK, 

ECCLES.  LANCASHIRE. 


CONTENTS 


CHAPTER  I 

ELEMENTARY   STUDY   OF   ALTERNATING   CURRENTS 

PACKS 

i.    Definition.  —  2.    Generation    of    Alternating    Currents.— 

3.  Graphical  Representation  of  an  Alternating  Current. — 

4.  Frequency.— 5.   Mean  and    \/Mean  Square  Values  of 
Alternating    Currents.  —  6.    Phase    Difference. — 7.    Sine 
Waves— Clock    Diagram.  —  8.    Addition    of    Alternating 
E.M.F.s.--  9.   Vector   Diagrams.— 10.  Addition  of  Alter- 
nating   E.M.F.s  with    the    aid  of    Vector    Diagrams.— 
ii.  Mean  Power  of  an  Alternating  Current. — 12.  Power 
Factor  ......  1—27 


CHAPTER  II 

SELF-INDUCTION   AND   CAPACITY 

13.  Magnetic  Field  due  to  a  Continuous  Current. — 14.  Mag- 
netic Field  due  to  Alternating  Current.— 15.  E.M.F. 
Produced  by  an  Alternating  Magnetic  Field. — 16.  Induct- 
ance.— 17.  Reactance. — 18.  Current  Flow  in  Circuit  of 
Negligible  Inductance. — 19.  Current  Flow  in  Circuit  of 
Appreciable  Inductance. — 20.  Diagram  showing  Relation 
of  E.M.F.s  in  Inductive  Circuit.— 21.  Effect  of  Iron  in 
Magnetic  Circuit.— 22.  Eddy  Currents. — 23.  Hysteresis.— 
24.  General  Conclusions  Regarding  the  Introduction  of 
Iron  in  the  Magnetic  Circuit. — 25.  Capacity. — 26.  Current 
Flow  through  a  Condenser.  —  27.  Condensive  React- 
ance -  -  -  28—64 
ix 


CONTENTS 


CHAPTER  III 

POLYPHASE  CURRENTS — GENERAL  PRINCIPLES  AND 
SYNCHRONOUS   GENERATORS 

PAGES 

28.  Polyphase  Currents. — 29.  Production  of  Rotary  Field. — 
30.  Three-Phase  Currents. — 31.  Utilization  of  Rotary 
Field.— 32.  Polyphase  Generators. — 33.  E.M.F.  Induced 
in  Generator  Windings. — 34.  Connections  of  Polyphase 
Armature  Windings.— 35.  Regulation  of  Synchronous 
Generators.— 36.  Compounding  .Synchronous  Generators. — 
37.  Parallel  Running  of  Alternators. — 38.  Synchronizers. — 
39.  Output  of  Polyphase  Generators  65 — 102 


CHAPTER  IV 

MEASUREMENT  AND  CALCULATION   OF  POWER   ON 
POLYPHASE   CIRCUITS 

40.  Power  Measurements. — 41.  Power  of  Two-Phase  Circuit.— 
42.  Power  in  Three- Phase  Circuit. — 43.  Vector  Diagram 
for  Calculating  Star  Resistances. — 44.  Power  Factor  of 
Three-Phase  Circuit. — 45.  Vector  Diagram  for  Balanced 
Inductive  Load. —46.  Unbalanced  Inductive  Load.— 
47.  Measurement  of  Power  in  High  Tension  Cir- 
cuits -.-..--  103—124 


CHAPTER  V 

POLYPHASE    TRANSFORMERS 

48.  Theory  of  Single-Phase  Transformer. — 49.  Vector  Diagram 
of  Transformer  without  Leakage.— 50.  Polyphase  Trans- 
formers.—51.  Methods  of  Connecting  Three-Phase  Trans- 
formers.—  52.  Efficiency  of  Transformers. — 53.  Phase 
Transformation  -  -  -  125 — 141 


CONTENTS  xi 

CHAPTER  VI 

POWER   TRANSMISSION    BY   POLYPHASE   CURRENTS 

PAGES 

54.  Power  Transmission  by  Polyphase  Currents. — 55.  Losses 
in  Transmission. — 56.  Choice  of  Voltage. — 57.  Trans- 
mission of  Single-Phase  Alternating  Currents. — 58.  Effect 
of  Inductive  Load  on  Line  Losses. — 59.  Effect  of  taking 
into  Account  the  Self-induction  of  the  Lines. — 60.  Pre- 
determination of  Inductive  Drop. — 61.  Capacity  of  Trans- 
mission Lines. — 62.  Vector  Diagram  for  Single-Phase 
Transmission  Line,  taking  into  Account  Resistance, 
Inductance,  and  Capacity.  — 63.  Rise  of  Pressure  at  the 
Distant  End  of  a  Long  Transmission  Line. — 64.  Trans- 
mission by  Two-Phase  Currents. — 65.  Transmission  by 
ThroPhase  Alternating  Currents. — 66.  Effect  of  In- 
ductive load  on  Line  Losses.— 67.  Self-induction  of  Three- 
Phase  Lines. — 68.  Capacity  of  Three-Phase  Transmission 
Lines. — 69.  Most  Economical  Section  of  Transmission 
Lines. — 70.  Relative  Advantages  of  Different  Systems  of 
Transmission. — 71.  Arrangement  of  Overhead  Lines  in 
Practice  -  -  142—192 


CHAPTER  VII 

POLYPHASE    INDUCTION   MOTORS 

72.  Introductory. — 73.  Magnetic  Leakage  in  Transformers. — 
74.  The  Induction  Motor  Considered  as  a  Special  Case  of 
the  Alternating  Current  Transformer. — 75.  Vector  Diagram 
for  Induction  Motor  with  Rotor  at  Rest. — 76.  Starting 
Torque  of  Polyphase  Induction  Motor.  —  77.  Vector 
Diagrams  for  Induction  Motor  with  Armature  in  Motion. — 
78.  Overload  Capacity  of  Induction  Motors  as  Influenced 
by  Magnetic  Leakage  and  Rotor  Resistance. — 79.  General 
Conclusions  Regarding  Magnetic  Leakage  in  Induction 
Motors. — 80.  Complete  Vector  Diagram  for  Polyphase 
Induction  Motor.— 81.  Efficiency  of  Polyphase  Motors. — 
82.  Methods  of  Measuring  the  Slip  of  Induction  Motors. — 


Xll  CONTENTS 


83.  Circle  Diagrams.— 84.  Circle  Diagram  Taking  Losses 
into  Account. — 85.  Methods  of  Starting  Induction  Motors. 
— 86.  Speed  Regulation. —87.  Reversing  Direction  of 
Revolution. — 88.  Recapitulation  -  -  -  193—246 


CHAPTER  VIII 

ASYNCHRONOUS  GENERATORS,  FREQUENCY  CONVERTERS, 

COMPENSATED  INDUCTION  MOTORS,  AND  ROTARY 

CONVERTERS 

89.  The  Polyphase  Induction  Motor  used  as  a  Generator. — 

90.  Vector    Diagrams    of    Asynchronous    Generator.  — 

91.  Conclusions  regarding  the  Induction  Motor  used  as  a 
Generator. — 92.  Frequency  Converters.— 93.  Compensated 
Polyphase  Motors  with   Commutators. — 94.  Synchronous 
Rotary  Converters.— 95.  Elementary  Theory  of  the  Rotary 
Converter.— 96.  Output   and    Efficiency   of  Rotary   Con- 
verters.—97.    Starting  and    Synchronizing    Rotary   Con- 
verters. —  98.    Regulation    of    Rotary    Converters.  —  99. 
Hunting    of    Rotary    Converters.  —  100.     Motor    Con- 
verters -  -  -      247 — 289 


APPENDIX  I 

ON  THE  RELATION  BETWEEN  MAGNETIC  FLUX  AND  INDUCED 
E.M.F.  IN  A  CIRCUIT  CONVEYING  AN  ALTERNATING 
CURRENT  ...  .  290 — 292 


APPENDIX  II 

METHOD  OF  DRAWING  THE  COMPLETE  VECTOR  DIAGRAM  FOR  A 
POLYPHASE  INDUCTION  MOTOR  FROM  THREE  SETS  OF 
MEASUREMENTS  MADE  ON  THE  MACHINE  -  -  292 — 296 

INDEX      - 297—300 


POLYPHASE   CURRENTS 

CHAPTER  I 

ELEMENTARY  STUDY  OF  ALTERNATING  CURRENTS 

As  an  introduction  to  the  subject  of  polyphase  alter- 
nating currents,  it  will  be  advisable  to  consider  the  general 
principles  which  underlie  the  working  of  all  alternating 
currents  of  electricity,  whether  single  -  phase  or  poly- 
phase ;  and  it  is  proposed  to  devote  this  and  the  follow- 
ing chapter  to  an  elementary  study  of  single-phase 
currents;  the  object  being  to  point  out  the  essential 
differences  between  direct  and  alternating  currents,  and 
explain  the  various  effects  peculiar  to  alternate-current 
working. 

i.  Definition. — An  alternating  current  is  —  as  indi- 
cated by  the  name — a  current  which,  instead  of  being 
unidirectional,  such  as  the  current  produced  by  a  battery 
or  a  continuous-current  dynamo,  flows  first  in  one  direc- 
tion and  then  in  the  opposite  direction  in  regular  succes- 
sion. In  other  words,  it  is  a  current  which,  starting  from 
zero  value,  increases  in  strength  in  a  positive  direction 
until  it  reaches  its  maximum  positive  value ;  then  dies 
down  to  zero  value,  after  which  it  rises  to  its  maximum 
negative  value,  and  again  falls  to  zero  ;  this  process  being 
repeated  in  a  periodic  manner. 

i 


2;'%;  :       .';         ALTERNATING  CURRENTS 

2.  Generation   of  Alternating  Currents.— There 
is  little   difference    in    principle    between   an    alternator 
and   a   direct-current    dynamo  ;    it    is    the    absence    of 
the   commutator    in   the   alternating  -  current    generator 
which  leads  to  the  current  obtained  from  the  terminals 
or  collecting  rings  being  of  the  character  described  above. 
Given  any  continuous  -  current  generator,  it   is  merely 
necessary  to  replace  the  commutator  by  a  couple  of  slip- 
rings  connected  to  suitable  points  on  the  armature  wind- 
ing, and  to  excite  the  field  coils  from  a  separate  continuous- 
current   source,   in    order    that   currents   alternating   in 
direction  may  be  drawn  from  the  armature. 

There  are  many  types  of  alternator  which  differ  little 
in  appearance  from  multipolar  direct-current  dynamos, 
and  it  will  readily  be  understood  that  the  number  of 
reversals  of  the  current  in  a  given  time  will  depend  upon 
the  speed  of  rotation  of  the  armature  and  the  number  of 
poles  in  the  exciting  field. 

3.  Graphical     Representation     of    an     Alter- 
nating Current. — Let  Fig.  i  represent   the   chart   of 
a  centre-zero  recording  ammeter,  such  as  might  be  used 
for  registering  the  charge  and  discharge  currents  of  a 
battery.     On  such  a  chart,  the  lapse  of  time  is  measured 
horizontally,  from  left  to  right,  while  the  strength  of  the 
current  is  indicated  by  vertical  distances  above  or  below 
the  horizontal  centre  line.     If  the  distance  is  measured 
above  this  horizontal  datum  line,  this  is  an  indication  that 
the  current  is  flowing  in  a  positive  direction,  corresponding, 
let  us  say,  with  a  charging  current ;  but  if  the  measure- 
ment is  made  below  the  datum  line,  this  indicates  that  the 
current  is  flowing  in  a  negative  direction,  and   that   the 
battery  is  being  discharged. 

Referring  to  the  curve  in  Fig.  i,  it  will  be  noticed  that, 
between  the  times  indicated  by  the  points  t  and  ^  on  the 


GRAPHICAL   REPRESENTATION 


datum  line,  corresponding  to  a  period  of  4  hours  and 
30  minutes,  the  charging  current  has  gradually  fallen 
from  50  amperes  down  to  25  amperes,  after  which  it  has 
increased  in  strength,  only  to  be  entirely  interrupted 
shortly  afterwards  before  gradually  increasing  again,  but 
this  time  in  a  negative  direction  —  i.e.,  as  a  discharging 
current. 

The  variation  of  an  alternating  current,  both  in  strength 
and  direction,  may  be  represented  in  a  similar  manner. 


(00- 


100- 


FIG.  i. 


In  such  a  curve  as  that  shown  in  Fig.  2,  the  lapse  of 
time  is  measured — as  in  the  previous  example — horizon- 
tally from  left  to  right,  while  the  strength  and  direction 
of  the  current,  at  any  particular  instant,  are  indicated  by 
the  length  of  the  vertical  ordinate  and  its  position  rela- 
tively to  the  datum  line.  The  chief  difference  between 
this  and  the  previous  diagram  (Fig.  i)  lies  in  the  fact 
that,  whereas  in  the  first  diagram  we  were  dealing  with 
a  lapse  of  time  measured  by  hours,  the  total  horizontal 


ALTERNATING  CURRENTS 


distance  in  Fig.  2  covers  only  a  small  fraction  of  a  second 
of  time ;  and,  further,  the  nature  of  the  curve,  instead  of 
being  irregular  as  in  Fig.  i,  is  now  such  that  each 
succeeding  positive  half-wave  is  exactly  similar  to  the 
preceding  one,  and  each  succeeding  negative  half-wave 
is  also  exactly  similar  to  those  that  came  before.  In  fact, 
the  current  we  are  now  considering  is  a  periodic  function 
of  the  time  :  the  current  rises  and  falls  through  a  certain 
cycle  of  values,  which  is  repeated  over  and  over  again  ; 


FIG.  2. 

and  the  periodic  time,  or  time  required  for  the  performance 
of  one  complete  cycle,  is  represented  in  Fig.  2  by  the 
horizontal  distance  O  b.  This  distance,  as  already  men- 
tioned, represents — in  the  case  of  the  currents  we  are  at 
present  concerned  with — only  a  small  fraction  of  a  second 
of  time  ;  and  what  is  known  as  the  periodicity  of  an  alter- 
nating current  is  the  number  of  complete  cycles  performed 
in  one  second. 

4.  Frequency. — The-  word  frequency,  when    used   in 
connection  with  alternating  currents,  is  synonymous  with 


MEAN  AND  MEAN  SQUARE  VALUES       5 

periodicity,  and  either  word  denotes  the  number  of  periods 
or  complete  cycles  per  second. 

Thus,  in  Fig.  2,  if  the  distance  O  b  is  equivalent  to 
i  -=-/  seconds,  the  periodicity  or  frequency  of  the  current 
represented  in  the  diagram  will  be  /,  and  the  number  of 
reversals  or  alternations  per  second  will  be  2  /.  In  practice 
the  number  of  periods  per  second  lies  usually  between 
25  and  100.  In  the  early  days  of  alternating  currents, 
when  these  were  used  solely  for  lighting  purposes,  the 
periodicity  was  commonly  about  100  in  this  country ; 
but  there  are  reasons  which  render  a  lower  frequency 
advisable,  and  from  80  to  25  is  the  usual  practice  nowa- 
days. In  America  the  standard  frequencies  are  60  and  25. 


5.  Mean  and  x  Mean  Square  Values  of  Alter- 
nating Currents.  —  Referring  again  to  Fig.  2,  it 
will  be  readily  understood  that  this  curve  may  stand 
for  any  alternating  quantity,  such  as  the  E.M.F.  of  a 
generator,  or  the  current  in  the  filament  of  an  incan- 
descent lamp ;  and,  in  any  case,  whether  the  curve 
represents  the  periodic  variation  of  amperes  or  volts,  the 
length  of  the  ordinate  C  B  is  a  measure  of  the  maximum 
positive  value  of  the  alternating  quantity,  and  the  distance 
C1B1  is  a  measure  of  the  maximum  negative  value. 

In  practice  the  negative  half- wave  is  generally  similar 
in  shape,  and  equal  in  magnitude,  to  the  positive  half- 
wave,  and  it  therefore  follows  that  the  ordinates  C  B  and 
ClBl  are  equal  in  length.  With  this  maximum  value  of 
an  alternating  quantity  we  shall  not  concern  ourselves  at 
present ;  it  will  suffice  to  point  out  that  this  value  of  an 
alternating  current  will  determine  the  total  magnetic  flux 
or  number  of  magnetic  lines  linked  through  the  circuit  at 
each  reversal ;  and  the  insulation  of  the  circuit  must  be 
considered  in  relation  to  the  maximum  value  of  the 
curve. 


O  ALTERNATING   CURRENTS 

With  respect  to  the  mean  or  average  value  of  an 
alternating  current,  it  might  be  supposed  that  this  is  the 
quantity  with  which  we  shall  be  principally  concerned ; 
but,  as  a  matter  of  fact,  this  is  not  the  case.  In  the 
design  of  alternators,  transformers,  electromagnetic 
measuring  instruments,  etc.,  it  is  undoubtedly  this  value 
of  the  induced  E.M.F.  which  is  most  easily  calculated; 
but  without  a  knowledge  of  the  shape  of  the  wave,  as 
shown  in  Fig.  2,  we  shall  still  be  without  some  most 
important  information.  It  should,  perhaps,  be  hardly 
necessary  to  explain  what  is  meant  by  the  mean  value  of 
.an  alternating  current  or  E.M.F.  :  we  have  merely  to 
take  the  average  of  all  the  ordinates  of  the  wave  diagram, 
or — what  amounts  to  the  same  thing — measure  the  area 
of  the  curve  O  B  d  (Fig.  2)  and  divide  by  the  length 
O  d.  If  a  planimeter  is  not  available,  recourse  can  be 
had  to  the  method  frequently  adopted  in  connection  with 
steam-engine  diagrams — that  is  to  say,  the  half-period  O  d 
would  be  divided  into  a  convenient  number  of  equal  parts, 
and  the  sum  of  the  lengths  of  all  ordinates  erected  on  the 
central  point  of  every  section,  when  divided  by  the  total 
number  of  such  ordinates,  will  approximate  to  the  true 
mean  value  of  the  current  or  E.M.F.,  as  the  case  may  be. 

Referring  still  to  Fig.  2 — which  we  shall  assume 
represents  the  variation  in  strength  and  direction  of  an 
alternating  current  of  electricity — let  us  suppose  that  the 
resistance  of  the  circuit  conveying  the  current  is  R  ohms; 
now,  if  I  is  the  value  of  the  current  at  any  instant,  the 
rate  at  which  work  is  being  done  in  heating  the  conductors 
will,  at  that  particular  instant,  be  equal  to  I2  R.  Hence, 
if  we  wish  to  know  the  average  rate  at  which  work  is  being 
done  by  an  alternating  current,  we  must  calculate  the 
mean  value  of  the  square  of  the  current  and  multiply  this 
quantity  by  the  resistance  R.  Thus,  instead  of  taking 


MEAN  AND  MEAN  SQUARE  VALUES       7 

the  mean  of  a  large  number  of  ordinates  of  the  half- wave 
O  B  d — as  in  the  previous  example — we  must  now  take 
the  average  of  the  sqttares  of  all  such  ordinates,  and  this 
quantity,  multiplied  by  the  ohmic  resistance  of  the  circuit, 
will  give  us  the  watts  lost  in  heating  the  conductors. 

It  follows,  therefore,  that  when  we  speak  of  an 
alternating  current  as  being  equal  to  a  certain  number  of 
amperes,  we  invariably  allude  to  that  value  of  the  current 
which,  when  squared  and  multiplied  by  the  resistance  of 
the  circuit  in  which  it  is  flowing,  will  give  us  the  actual 
power  in  watts  which  is  being  spent  in  overcoming  the 
resistance  of  the  conductors. 

Thus  it  is  the  root-of-the-mean-sqitare  value  of  an 
alternating  current  which,  as  far  as  power  measurements 
are  concerned,  enables  us  directly  to  compare  a  periodically 
varying  current  with  a  continuous  current  of  constant 
strength :  it  is  the  product  of  this  value  of  the  current 
and  the  corresponding  value  of  the  resultant  E.M.F.  to 
which  it  owes  its  existence  which,  in  all  cases,  is  a 
measure  of  the  power  absorbed  in  the  circuit. 

The  readings  of  nearly  all  commercial  measuring 
instruments  for  alternating  currents  depend  upon  the 
^mean  square  value  of  the  current  or  E.M.F. ;  and  it  is 
only  in  exceptional  cases  that  we  require  to  know 
either  the  maximum  or  the  true  mean  value  of  an  alternating 
quantity. 

If  we  apply  a  potential  difference  of  100  volts  to  the 
terminals  of  an  incandescent  lamp — the  inductance  of 
which  is  very  small,  and  practically  negligible* — the 
lamp  will  glow  with  the  same  brilliance  whether  this 
potential  difference  is  obtained  from  a  continuous  or  an 

*  The  great  importance  of  the  magnetic  induction  in  a  circuit 
conveying  an  alternating  current  will  be  fully  dealt  with  in  the 
following  chapter. 


8 


ALTERNATING  CURRENTS 


alternating  current  source  ;  and  when  we  speak  of  an 
alternating  E.M.F.  of  i  volt,  it  is  the  ^/mean  square 
value  of  the  alternating  voltage  to  which  we  refer.  Again, 
when  we  speak  of  an  alternating  current  of,  say, 
10  amperes,  we  invariably  refer  (unless  special  mention 
is  made  to  the  contrary)  to  the  N/mean  square  value  of 
the  periodic  current  which,  so  far  as  heating  effects  are 


FIG.  3. 


concerned,  is  exactly  equivalent  to  a  continuous  current  of 
10  amperes.  This  value  of  an  alternating  quantity  is 
known  as  the  virtual  or  effective  value. 

6.  Phase  Difference. — Two  alternating  E.M.F.s  of 
the  same  character  and  periodicity,  or  an  alternating 
current  and  the  E.M.F.  to  which  it  owes  its  existence, 
are  said  to  be  in  phase  when  the  growth  and  decrease, 


PHASE    DIFFERENCE  9 

reversal,  and  maximum  values  (of  the  same  sign)  occur 
simultaneously. 

Thus,  in  Fig.  3,  let  the  full-line  curve  I  represent  the 
periodic  variations  of  the  current  flowing  in  a  given 
circuit,  and  let  the  dotted  curve  E  represent  the  alternating 
E.M.F.  in  the  circuit :  the  periodicity  of  these  two 


FIG.  4. 

quantities  is  evidently  the  same,  as  indicated  by  the 
length  of  the  line  O  t — representing  the  time  of  one 
complete  period — being  equal  in  both  cases.  Moreover, 
it  will  be  observed  that  the  current  and  E.M.F.  pass 
through  zero  value,  reach  their  maximum  positive  value, 
reverse  their  direction,  and  pass  through  their  maximum 


10  ALTERNATING   CURRENTS 

negative  value  at  precisely  the  same  instant  of  time,  and 
they  are  therefore  in  phase. 

Consider,  now,  the  two  curves  drawn  in  Fig.  4. 

We  shall  assume  that  these  represent  two  E.M.F.s  of 
the  same  wave  shape  which,  for  the  sake  of  the  argu- 
ment, might  be  produced  by  two  alternators,  similar  in 
all  respects,  and  having  their  shafts  rigidly  coupled 
together. 

These  two  E.M.F.s  are  of  the  same  periodicity — the 
distances  t  and  tl  being  equal — but  in  this  case  the  two 
alternating  quantities  are  no  longer  in  phase.  It  will  be 
noted  that,  throughout  the  complete  cycle,  the  instan- 
taneous values  of  the  E.M.F.  represented  by  the  curve 
E!  occur  exactly  one-eighth  of  a  period  after  the  corre- 
sponding values  of  the  curve  E  ;  and  Ex  is  therefore  not 
in  phase  with  E,  but  lags  behind  this  E.M.F.  by  a  small 
fraction  of  a  second  represented  by  the  distance  d,  which, 
in  this  example,  is  exactly  equal  to  one-eighth  of  a  com- 
plete period.  In  other  words,  there  is  a  difference  of  phase 
between  these  two  E.M.F.s  equal  to  one-eighth  of  a 
period. 

7.  Sine  Waves  —  Clock  Diagram.  —  In  the  last 
article,  the  component  waves,  E  and  Ex  of  Fig.  4,  were 
assumed  to  be  of  the  same  shape,  and,  as  a  matter  of 
fact,  they  are  actually  sine  waves,  representing  a  simple 
periodic  variation  of  the  E.M.F.  This  being  the  form  of 
wave  that  must  necessarily  be  assumed  in  all  mathe- 
matical methods  of  solving  alternating  current  problems, 
it  is  well  that  the  reader  should  understand  what  is 
involved  in  this  assumption. 

Consider  the  so-called  clock  diagram  of  Fig.  5,  in  which 
the  length  of  the  line  O  B  is  a  measure  of  the  maximum 
value  of  the  alternating  current  or  E.M.F.  (C  B  or 
Cx  B!  in  Fig.  2),  and  suppose  the  line  O  B  to  revolve 


SINE   WAVES 


II 


round  the  point  O  in  the  direction  indicated  by  the  arrow. 
If,  now,  we  consider  the  projection  of  this  revolving 
line  upon  any  fixed  line — such  as  the  vertical  diameter 
M  N  of  the  dotted  circle — it  will  be  seen  that  the  speed 
of  O  B  can  be  so  regulated  that  the  length  of  this  projec- 
tion will,  at  any  moment,  be  a  measure  of  the  instan- 
taneous value  of  the  variable  current  or  E.M.F.  It  will 


FIG.  5. 

also  be  evident  that — since  the  alternating  quantity  must 
pass  twice  through  its  maximum  value,  and  twice  through 
zero  value,  in  the  time  of  one  complete  period — the  line 
O  B  must,  in  all  cases,  perform  one  complete  revolution 
in  i//  seconds  ;  where  /  is  the  frequency,  or  number  of 
periods  per  second.  Also,  in  order  that  this  diagram 
may  give  us  all  the  information  needed,  it  will  be  con- 
venient to  assume  that  all  measurements,  such  as  O  d, 


12  ALTERNATING   CURRENTS 

which  are  made  above  the  line  P  Q,  refer  to  positive  values 
of  the  variable  quantity,  whereas  all  measurements  made 
below  will  apply  to  the  negative  values. 

When  the  line  O  B  (Fig.  5)  is  vertical,  its  projection 
O  d  is  equal  to  it ;  we  therefore  conclude  that  the  alter- 
nating quantity  is  at  that  moment  passing  through  its 
maximum  positive  value.  As  O  B  continues  to  move 
round  in  a  clockwise  direction,  O  d  will  diminish,  until 
the  point  B  has  moved  to  Q,  when  O  d  will  be  zero; 
after  which  it  will  again  increase  in  length,  but  this  time 
— since  it  is  now  below  the  line  P  Q — the  flow  of  current 
or  the  direction  of  the  E.M.F.  is  reversed.  At  N,  the 
maximum  negative  value  will  be  reached,  only  to  fall  again 
to  zero  at  P  ;  after  which  it  rises  once  more  to  the 
positive  maximum  at  M. 

If  the  line  O  B  revolves  round  O  at  a  uniform  rate,  the 
point  d  will  move  to  and  from  the  centre  O  with  a  simple 
periodic  or  simple  harmonic  motion.  It  follows  that,  if 
the  length  of  the  projection  O  d  represents  the  variations 
of  an  alternating  E.M.F.,  this  E.M.F.  must  be  under- 
stood to  be  rising  and  falling  in  a  simple  periodic  manner  ; 
and  since  the  length  O  d  will  now  be  proportional  to  the 
sine  of  the  time  angle  0,  the  shape  of  the  wave  (Fig.  4) 
will  be  that  of  a  curve  of  sines,  the  characteristic  feature 
of  which  is  that  every  ordinate  such  as  h  e  will  be  propor- 
tional to  the  sine  of  its  horizontal  distance  from  O  ;  this 
distance  being  now  measured,  not  in  time,  but  in  angular 
measure,  it  being  understood  that  360  degrees  correspond 
to  the  time  of  one  complete  period. 

The  expression  "  phase  difference  "  or  "  angle  of  lag  " 
has  no  definite  meaning  when  applied  to  waves  of 
irregular  shape :  the  lapse  of  time  between  successive 
maximum  values  may  not  be  the  same  as  the  interval 
between  successive  zero  values  of  the  quantities  compared  ; 


ADDITION    OF   ALTERNATING   E.M.F.S  13 

and  in  the  treatment  of  alternating-current  problems,  it  is 
therefore  usual  to  assume  that  the  wave  shapes  are  either 
actually  sine  curves,  or  that  so-called  "equivalent  sine 
waves  "  have  been  substituted  for  the  actual  waves. 

8.  Addition  of  Alternating  E.M.F.s. — Let  us  sup- 
pose that  the  two  alternators,  with  shafts  rigidly  coupled 
together,  generating  two  alternating  E.M.F.s  with  a  phase 
difference  of  one-eighth  of  a  period,  are  electrically  joined 
in  series  on  the  same  circuit ;  and  let  us  consider  what 
will  be  the  resultant  E.M.F.  in  the  circuit. 

When  dealing  with  continuous  currents,  it  is  merely 
necessary  to  add  together  the  various  E.M.F.s — such  as 
those  due  to  a  dynamo  in  series  with  a  battery — paying 
due  regard  to  their  respective  signs— i.e.,  whether  they 
are  acting  in  a  positive  or  a  negative  direction — in  order 
to  obtain  the  resultant  E.M.F.  producing,  or  tending  to 
produce,  a  flow  of  current  in  the  circuit.  And  so  also  in 
the  case  of  alternating  currents,  at  any  particular  instant  of 
time  the  resultant  E.M.F.  in  any  circuit  is  equal  to  the 
algebraical  sum  of  the  various  E.M.F.s  in  the  circuit. 
Thus,  on  referring  again  to  Fig.  4,  it  will  be  noticed  that 
a  dotted  curve  E2  has  been  drawn  in  addition  to  the  two 
curves  E  and  Er  Every  ordinate  of  this  curve,  such  as 
h  e2,  is  equal  to  the  sum  of  the  ordinates,  h  e  and  h  el  of  the 
two  full-line  curves ;  due  attention  being  paid  to  the  sign 
of  these  instantaneous  values  of  the  E.M.F. 

For  instance,  when  the  positive  value  of  Ex  is  exactly 
equal  to  the  negative  value  of  E,  the  resultant  E.M.F. 
in  the  circuit  will  be  nil,  and  this  is  what  occurs  at  the 
instant  p  where  the  resultant  curve,  E2,  passes  through 
its  zero  value. 

A  cursory  examination  of  this  diagram  (Fig.  4)  will 
make  it  clear  that  the  resultant  E.M.F.,  E2,  is  of  the 
same  periodicity  as  the  two  component  E.M.F.s;  and 


14  ALTERNATING   CURRENTS 

not  only  its  maximum  value,  but  also  its  virtual  value, 
will  be  something  less  than  would  be  obtained  by  merely 
adding  together  the  corresponding  values  of  the  E.M.F.s 
E  and  Er* 

9.  Vector  Diagrams. — The  method  described  above 
for  graphically  adding  together  two  (or  more)  periodically 
alternating  quantities  is  very  tedious  and  altogether  un- 
practical. The  method  about  to  be  described — which 
involves  the  proper  understanding  of  a  vector  quantity — 
is  of  such  general  utility  in  the  solution  of  alternating- 
current  problems  that  the  reader  who  may  not  be  familiar 
with  these  diagrams  should  devote  his  most  careful  atten- 
tion to  the  following  explanations  : 

In  the  first  place,  a  vector  quantity  (as  distinguished 
from  a  scalar  quantity)  possesses  not  only  magnitude,  but 
also  direction. 

A  magnitude  can  be  graphically  represented  by  the 
length  of  a  straight  line,  and  a  direction  can  be  represented 
by  the  angle  which  a  line  (of  indefinite  length)  makes 
with  a  datum  line  drawn  on  the  same  plan  for  the  purpose 
of  reference ;  but  a  vector  quantity  can  only  be  graphically 
represented  by  a  line  of  definite  length  drawn  in  a  definite 
direction  relatively  to  some  datum  line. 

The  resultant  magnetic  force  at  any  particular  point 
in  space  is  a  vector  quantity,  because  this  force  has  not 
only  a  definite  direction,  but  it  has  also  a  definite  intensity ; 
and  if  it  were  customary  to  indicate,  on  geographical  maps, 
the  magnetic  condition  at  any  given  place,  a  line  would 
have  to  be  drawn  not  only  in  a  certain  direction,  but  also 

*  It  is  only  when  the  two  component  waves  are  similar  in  shape, 
and  in  phase,  that  the  resultant  E.M.F.  will  be  exactly  equal— both 
in  regard  to  its  maximum  and  its  \/mean  square  values— to  the 
simple  addition  of  the  corresponding  values  of  the  component 


VECTOR   DIAGRAMS  15 

of  a  definite  length  such  as  to  indicate,  let  us  say,  the 
intensity  of  the  horizontal  component  of  the  earth's 
magnetism  at  that  place.  Such  a  line,  drawn  to  scale, 
with  an  arrow-head  at  one  end  (to  indicate  direction) 
would  be  what  is  commonly  known  as  a  vector. 

Instead  of  representing  an  alternating  current  or  E.M.F. 
by  means  of  the  wave  diagrams  with  which  we  have  been 
studying  them  up  to  the  present,  let  us  assume  that  we  are 
in  no  wise  concerned  with  the  maximum  or  mean  values, 
or  the  law  of  variation  of  an  alternating  quantity,  but 
only  with  its  ^/mean  square  or  virtual  value,  which,  as 
explained  above,  is  what  we  most  frequently  wish  to 
know.  This  is  also  the  quantity  which  is  measured  by 
an  alternating-current  ammeter  or  voltmeter.  A  straight 
line  of  such  a  length  as  to  indicate  the  amount  or  intensity 
of  this  alternating  quantity  drawn  in  any  direction  on  a 
sheet  of  paper  would  be  a  correct  graphical  representa- 
tion of  a  definite  number  of  amperes  or  volts.  But 
when  dealing  with  two  or  more  alternating  forces  all 
acting  in  the  same  circuit,  it  is  necessary  that  the  straight 
lines  representing  these  forces  should  be  not  only  of 
definite  lengths ;  they  must  also  be  drawn  at  certain 
definite  angles  relatively  to  each  other  in  order  to  repre- 
sent the  phase  differences  between  them. 

In  Fig.  6  the  complete  circle  of  360  degrees  (or  2  TT 
radians)  represents  one  period,  and  in  order  to  indicate 
the  phase  difference  between  the  alternating  quantities 
E  and  I,  two  straight  lines  OE  and  O  I — starting  from 
the  common  point  O,  and  of  such  lengths  as  to  indicate 
the  respective  magnitudes  of  these  quantities — are  drawn 
with  an  angle  0  between  them  such  that  the  ratio  which 
this  angle  bears  to  the  complete  circle  expresses  the 
phase  difference  between  E  and  I  as  a  fraction  of  the 
complete  period. 


i6 


ALTERNATING  CURRENTS 


For  instance,  if  the  angle  6  is  equal  to  45  degrees, 

7T 

or  -  radians  (as  in  Fig.  6),  this  means  that  the  current  I 

lags  behind  the  E.M.F.  E  by  a  time  interval  equal  to  one- 
eighth  of  a  period;  or,  in  other  words,  the  E.M.F.  E  is 
in  advance  of  the  current  I  by  this  same  interval  of  time. 
Thus,  if  the  periodicity  of  this  particular  E.M.F.  is  40? 


FIG.  6. 

it  follows  that  the  angle  I  O  E  in  Fig.  6  represents  a  time 
interval  of  ^J^th  of  a  second. 

It  is  not  usual  to  express  a  phase  difference  as  a  definite 
interval  of  time,  because  this  would  convey  no  useful 
information,  unless  the  periodicity  were  also  stated. 
What  we  wish  to  know  is  the  fraction  of  a  complete 
period  by  which  one  of  the  alternating  quantities  is  in 
advance  of  the  other,  and  this,  as  explained  above,  can 


ADDITION    OF   ALTERNATING   E.M.F.S  I/ 

readily  be  stated  as  an  angle ;  it  is  only  necessary  to  bear 
in  mind  that  the  complete  circle  (360  degrees)  stands  for 
one  period  or  double  alternation. 

If  the  reader  has  carefully  followed  what  has  been 
said  regarding  the  graphic  representation  of  alternating 
quantities  by  means  of  vectors,  it  is  more  than  probable 
that  he  may  yet  experience  considerable  difficulty  in 
forming  a  clear  conception  of  the  phase  difference — not 
between  the  respective  maximum  or  zero  values,  or  any 
other  definite  instantaneous  value  of  two  alternating 
quantities,  but  between  their  average  or,  what  is  more 
important,  their  *Jmean  square  values. 

This  difficulty  is  considered  in  the  following  article. 

10.  Addition  of  Alternating  E.M.F.s  with  the 
Aid  of  Vector  Diagrams.— Consider  once  again  (as 
in  articles  6  and  8)  two  alternators,  A  and  B,  joined  in 
series :  and  assume  that  they  are  similar  in  all  respects, 
with  the  same  number  of  poles,  and  are  driven  at  the 
same  speed. 

Let  us  suppose  three  voltmeters  to  be  connected  as 
shown  in  Fig.  7.  These  voltmeters  must  be  such  as  may 
be  used  indifferently  on  alternating  or  direct  current  cir- 
cuits ;  that  is  to  say,  they  must  measure  the  Vmean  square 
values  of  the  alternating  volts. 

The  voltmeters  Et  and  E2  will  indicate  the  volts  due 
respectively  to  the  alternators  A  and  B,  whereas  E  will 
measure  the  resultant  volts  at  terminals,  If  the  volts 
measured  by  E  are  equal  to  the  arithmetic  sum  of  the 
volts  Ex  and  E2,  the  two  machines  would  be  said  to  be 
in  phase ;  but,  as  a  rule,  the  reading  on  E  will  be  smaller 
than  the  arithmetic  sum  of  Et  and  E2.  We  will  suppose 
these  three  values  to  be  known.  From  the  centre  O 
(Fig.  8)  describe  a  circle  of  radius  O  E,  the  length  of 


iS 


ALTERNATING   CURRENTS 


which  is  a  measure  of  the  volts  E.  Now  draw  O  Ex  in 
any  direction  to  represent  the  volts  Er  From  Et  as  a 
centre  describe  an  arc  of  radius  Ex  E,  the  length  of  which 
is  proportional  to  the  volts  E2 :  it  will  cut  the  arc  already 
drawn  at  the  point  E.  Join  O  E,  and  complete  the 
parallelogram  O  Ex  E  E2.  The  angle  0  between  the  two 
component  vectors  O  Ex  and  O  E2  is  a  measure  of  what 
we  must  now  understand  as  the  angle  of  lag  or  phase 
difference  between  two  alternating  quantities  which  are  of 
the  same  frequency  and  wave  shape. 


FIG.  7. 

The  question  of  compounding  two  or  more  alternating 
forces  in  an  electric  circuit  now  becomes  a  very  simple 
matter.  Thus,  in  Fig.  8,  had  we  been  given  the  two 
voltages  El  and  E2  and  the  phase  difference  9  (instead  of 
the  three  voltages),  we  could  have  calculated  the  total 
E.M.F.,  E,  and  ascertained  its  phase  relation  to  the  two 
component  forces,  by  merely  constructing  the  triangle  of 
forces  O  Ex  E  in  the  manner  familiar  to  every  engineer. 

As  an  example,  let  us  suppose  that  there  are  three 
distinct  alternating  E.M.F.s — A,  B,  and  C — of  the  follow- 
ing values,  all  combining  to  produce  one  resultant  E.M.F. 
in  an  electric  circuit : 

A  =  200  volts. 

B  =  150  volts. 

C  =  100  volts. 


ADDITION    OF   ALTERNATING   E.M.F.S  19 

We  shall  also  assume  that  B  lags  behind  A  by  exactly  a 
quarter  of  a  period  (90  degrees),  while  C  leads,  or  is  in 
advance  of  A,  by  the  fraction  of  a  period  denoted  by  an 
angle  of  35  degrees. 

Draw  the  three  vectors  O  A,  O  B,  and  O  C,  in  Fig.  9,  to 
a  suitable  scale,  and  in  such  directions  that  the  angles 
A  O  B  and  A  O  C  are  respectively  equal  to  90  degrees  and 
35  degrees,  bearing  in  mind  that  O  B  must  be  drawn 


\ 
\ 

\ 


\ 
\ 

\ 
\ 


FIG.  8. 

behind  O  A,  while  O  C  must  be  drawn  in  advance*  Con- 
struct the  polygon  of  forces  O  B  r  R  closed  by  the  resultant 
OR,  which  represents  the  sum  of  the  three  forces — or 
vectors — O  A,  O  B,  O  C,  and  which  lags  behind  the  vector 

*  In  accordance  with  the  convention  adopted  by  international 
agreement,  the  counter-clockwise  direction  indicates  advance  in  phase. 
In  this  respect  the  vector  diagrams  differ  from  those  in  the  previous 
edition  of  this  book. 


20 


ALTERNATING   CURRENTS 


O  A  by  an  angle  0  equal,  in  this  example,  to  18  degrees. 
The  length  of  the  line  O  R,  measured  to  the  same  scale 
as  the  three  component  vectors,  gives  us  the  value  (in 
volts)  of  the  resultant  E.M.F.  In  this  example  it  will 
be  found  to  be  297  volts. 

Although  the  diagram  Fig.  9  has  been  drawn  to  scale, 
and  the  problem  solved  by  taking  actual  measurements 


90< 


35C 


FIG.  9. 

off  the  diagram,  it  should  be  stated  that  this  method  will 
usually  be  found  unpractical,  mainly  because  of  great 
differences  in  the  magnitudes  of  the  quantities  dealt  with, 
and  the  difficulty  experienced  in  locating  the  exact  point 
of  intersection  of  two  lines  when  the  angle  between  them 
is  small.  It  will  generally  be  found  convenient  to  draw 
the  vector  diagram  (not  necessarily  to  scale),  and  then 


MEAN    POWER  21 

solve  for  the  unknown  quantities  by  using  trigonometrical 
formulas  or  vector  algebra.  When  using  the  algebraic 
method,  every  vector  is  resolved  into  its  vertical  and  hori- 
zontal components,  and  the  various  operations  are  per- 
formed on  these  components  only.  The  addition  of 
several  alternating  currents  of  the  same  frequency 
therefore  resolves  itself  into  the  process  of  finding  the 
resultant  of  several  vectors,  much  as  a  surveyor  would 
compute  the  result  of  a  day's  traverse  by  "  latitude  and 
departure." 

n.  Mean  Power  of  an  Alternating  Current. — 

In  article  8  (p.  13),  the  two  full-line  curves  in  Fig.  4 
were  supposed  to  represent  two  E.M.F.s,  and  it  was 
shown  how  these  could  be  added  together  to  produce 
the  resultant  curve  E2.  Let  us  now  consider,  not  two 
E.M.F.s  differing  in  phase,  but  an  E.M.F.  and  a  current, 
also  differing  in  phase  ;  and  let  us  multiply  one  by  the 
other  in  order  to  obtain  the  power  in  the  circuit. 

From  what  has  been  said  on  the  subject  of  phase 
differences  generally,  the  reader  should  have  no  difficulty 
in  understanding  that,  whereas  in  the  case  of  continuous 
currents  the  power  in  a  circuit  is  given  by  the  product  of 
terminal  potential  difference  and  total  current,  this  is 
very  rarely  true  in  the  case  of  alternating  currents. 

If  we  connect  a  voltmeter  across  the  terminals  of  an 
alternating-current  circuit,  and  multiply  the  reading  on 
this  voltmeter  by  the  actual  (virtual)  value  of  the  current 
in  amperes,  we  shall  obtain  a  number  representing  what 
is  sometimes  called  the  apparent  power ;  but  this  will  not 
necessarily  be  a  measure  of  the  power  actually  being 
supplied  to  the  circuit.  The  true  power  supplied  to  the 
circuit  at  any  moment  will  be  given  by  the  product  of  the 
instantaneous  values  of  current  and  impressed  potential 
difference,  and  the  mean  value  of  all  such  products,  taken 


22 


ALTERNATING  CURRENTS 


during  the  time  of  one  complete  period,  will  be  the 
quantity  which  we  require  to  know. 

In  Fig.  10  the  power — or  watt — curve  has  been  drawn; 
it  is  obtained  by  multiplying  together  the  corresponding 
ordinates  of  the  curves  of  impressed  potential  difference 
E  and  of  the  current  I. 

Since  the  current  lags  behind  the  potential  difference, 
it  follows  that,  during  certain  portions  of  the  complete 


FIG.  10. 

period,  the  simultaneous  values  of  E  and  I  will  be  of 
opposite  sign ;  that  is  to  say,  the  current  will  be  flowing 
against  the  impressed  E.M.F. :  the  work  done  will  there- 
fore be  negative,  and  these  ordinates  of  the  watt  curve 
will  have  to  be  plotted  below  the  datum  line.  This 
negative  work  (which  is  equal  to  the  area  of  the  shaded 
curve  below  the  datum  line)  may  sometimes  almost  equal 


MEAN    POWER  23 

the  positive  amount  of  work  done,  in  which  case  the 
current  is  practically  wattless— i.e.,  the  amount  of  energy 
put  into  the  circuit  during  one  quarter  period  is  given 
back  again  during  the  succeeding  quarter  period. 

(The  proper  understanding  of  this  state  of  things  must 
of  necessity  present  some  difficulties  to  those  unacquainted 
with  alternating  currents ;  but  when  dealing  with  the  re- 
actance of  an  alternating-current  circuit,  the  matter  will 
be  again  referred  to.) 

Returning  to  a  consideration  of  the  curves  of  Fig.  10, 
we  see  that  the  total  amount  of  work  done  during  one 
complete  period  is  equal  to  the  area  of  the  two  shaded 
curved  curves  marked  + ,  less  the  area  of  the  two  shaded 
curves  marked  —  ;  and  the  average  power  supplied  to  the 
terminals  of  the  circuit  will,  therefore,  be  indicated  by 
the  average  ordinate  of  this  dotted  watt  curve,  due  atten- 
tion being  paid  to  the  sign  of  the  instantaneous  power 
values. 

Instead  of  drawing  a  diagram  such  as  Fig.  10,  which 
is  a  very  laborious  undertaking,  let  us  see  what  can  be 
done  with  the  simpler  and  more  convenient  vector 
diagrams. 

In  Fig.  ii  the  Vmean  square  value  of  the  E.M.F.  is 
represented  by  the  vector  O  E,  and  the  x/mean  square 
value  of  the  current  by  O  I. 

The  phase  difference  between  these  two  alternating 
quantities  is  indicated  by  the  angle  6 ;  and  since  O  E  is 
in  advance  of  O  I  (refer  to  footnote  on  p.  19),  it  follows 
that  the  current  I  lags  behind  the  E.M.F.,  E. 

The  method  of  compounding  or  adding  together  two 
or  more  E.M.F.s  was  explained  in  article  10  (p.  17);  and 
it  follows  from  this  that  any  vector,  such  as  O  E,  repre- 
senting an  E.M.F.,  can  be  considered  as  being  made  up 
of  two  or  more  component  vectors. 


24  ALTERNATING  CURRENTS 

Thus,  in  Fig.  n,  if  we  draw  through  O  the  line  O  B  at 
right  angles  to  O  I,  and  from  the  point  E  drop  perpen- 
diculars E  e  and  E  el  on  to  these  two  lines,  we  have  in  O  e 
and  O  el  two  vectors  representing  imaginary  E.M.F.s 
which,  when  added  together  in  the  manner  explained  in 
article  10,  would  produce  the  resultant  E.M.F.,  E. 

..A 


FIG.  ii. 

Suppose,  now,  that  there  is  no  other  E.M.F.  in  the 
circuit  but  that  represented  by  the  vector  O  e.  This  is 
in  phase  with  the  current,  and  the  true  watts  are  there- 
fore obtained  by  multiplying  together  this  E.M.F.,  -et  and 
the  current  I.* 

*  See  article  5,  especially  the  concluding  paragraphs  (p.  7). 


MEAN    POWER  2$ 

If,  on  the  other  hand,  e1  were  the  only  E.M.F.  in  the 
circuit,  the  current  I  would  be  wattless — i.e.t  the  mean 
power  in  the  circuit  would  be  nil — because,  when  the 
phase  difference  between  the  current  and  E.M.F.  is 
90  degrees,  the  energy  put  into  the  circuit  during  one 
quarter  of  a  period  is  given  back  again  during  the  next 
quarter  of  a  period,  and  the  mean  watts  are,  therefore, 
equal  to  zero. 

This  leads  us  to  the  conclusion  that,  of  the  two 
(imaginary)  components,  e  and  elt  of  the  total  E.M.F.,  it 
is  only  the  component  et  in  phase  with  the  current,  which 
need  be  taken  into  account  when  calculating  the  power 
in  the  circuit;  and,  given  the  angle  of  lag  (or  phase 
difference)  between  the  applied  potential  difference  E 
and  current  I,  the  total  power  in  the  circuit  is  obtained 
by  projecting  one  of  the  vectors — such  as  O  E — upon  the 
other — O  I — and  then  multiplying  this  latter  quantity  by 
the  projection  (O  e)  of  the  first  one  upon  it. 

In  order  to  obtain  a  graphical  representation  of  the 
power  supplied  to  a  circuit  in  which  the  current  I  lags 
behind  the  impressed  volts  E  by  an  amount  equal  to  the 
angle  0,  we  have  simply  to  move  round  one  of  the  vectors, 
let  us  say  O  I,  through  an  angle  of  90  degrees,  and  then 
construct  the  parallelogram  O  A,  the  area  of  which  will  be 
a  measure  of  the  average  value  of  the  true  watts ;  for  it 
is  evident  that  this  area  will  always  be  equal  to  OlxOe. 

12.  Power  Factor.— Since  e  O  E  (Fig.  n)  is  a  right- 
angled  triangle,  it  follows  that  the  power  (Ixe)  supplied 
to  the  circuit  can  be  written  I  x  E  cos  0.  But  I  x  E  is 
what  we  have  already  called  the  apparent  watts.  Hence 
the  real  watts  =  the  apparent  watts  x  cos  0,  and  this  is  the 
definition  of  the  angle  of  lag  between  current  and  im- 
pressed E.M.F.,  which  is  the  most  useful.  It  is  this 
multiplier  (cos  0)  to  which  the  name  power  factor  has  been 


26  ALTERNATING   CURRENTS 

given.  The  power  factor  of  a  circuit-carrying  current 
under  definite  conditions  is  the  ratio  of  the  true  power 
to  the  apparent  power  or  volt -amperes.  It  cannot  be 
greater  than  unity,  and,  in  a  circuit  carrying  an  alternating 
current,  is  usually  less  than  unity.* 

As  this  expression  occurs  frequently  in  all  literature 
dealing  with  alternating  currents,  it  is  important  that 
the  reader  should  have  a  clear  understanding  of  its 
meaning. 

In  connection  with  continuous  currents,  the  term  is  not 
used,  for  the  simple  reason  that,  once  a  steady  flow  of 
current  has  been  established,  the  value  of  this  current  is 
always  equal  to  the  ratio  of  applied  E.M.F.  to  ohmic 
resistance  of  the  circuit;  and  the  loss  of  power  in  the 
circuit  may  either  be  written  P  R  or  E  I. 

In  the  case  of  alternating  currents,  it  very  often  happens 
that  the  current  is  not  in  phase  with  the  applied  E.M.F., 
and  the  causes  leading  to  this  displacement  of  phase  are 
sufficiently  important  to  justify  our  devoting  the  following 
chapter  to  their  consideration.  The  effects  of  such  phase 
displacement  between  current  and  E.M.F.  have,  however, 
already  been  explained,  and  it  will  be  understood  that, 
although  the  power  spent  in  heating  the  conductors  of 
the  circuit  may  still  be  written  PR,  the  product  E  x  I 
only  represents  the  true  power  when  the  power  factor  is 

*  It  has  been  assumed  that  the  reader  is  not  without  at  least 
an  elementary  knowledge  of  trigonometry  ;  but  should  this  assump- 
tion be  unwarranted,  it  will  suffice  to  point  out  that  the  cosine 
of  the  angle  Q  is  the  trigonometrical  function  of  the  angle, 
the  numerical  value  of  which  is  always  obtained  by  dividing 
the  length  of  the  side  O  e  of  the  right-angled  triangle  E  O  e 
by  the  length  of  the  hypotenuse,  O  E.  The  expression  cos  9  must, 
therefore,  be  understood  to  indicate  the  ratio  of  O  e  to  O  E,  which 
ratio  will,  obviously,  be  constant  for  a  given  angle,  and  quite 
independent  of  the  actual  lengths  O  E  and  O  e. 


POWER   FACTOR  27 

unity — i.e.,  when  the  current  is  in  phase  with  the  applied 
E.M.F.  Whenever  there  is  the  slightest  phase  displace- 
ment between  the  applied  E.M.F.  and  the  resulting 
current,  the  apparent  power  (E  x  I)  has  to  be  multiplied  by 
the  power  factor  (cos  0)  in  order  to  arrive  at  the  true  power 
in  the  circuit. 


CHAPTER  II 

SELF-INDUCTION    AND    CAPACITY 

13.  Magnetic  Field  Due  to  a  Continuous 
Current. — Wherever  there  is  a  current  of  electricity 
there  must,  of  necessity,  be  a  corresponding  magnetic 
condition  of  the  surrounding  medium. 

In  the  case  of  continuous  currents,  once  a  steady  value 
has  been  reached,  the  magnetic  condition — i.e.,  the  number 
and  direction  of  the  magnetic  lines  in  the  circuit — remains 
unaltered.  Thus,  so  long  as  the  resultant  exciting 
ampere-turns  in  a  dynamo  machine  or  direct-current  motor 
remain  constant,  the  total  magnetic  flux  is  maintained  at 
a  definite  steady  value.  In  order  to  calculate  this  total 
flux,  it  is  necessary  to  know,  not  only  the  current  and  the 
number  of  turns  in  the  exciting  coils,  but  also  the 
dimensions  and  configuration  of  the  magnetic  circuit 
which  is  linked  with  the  electric  circuit,  and  especially 
the  amount  and  disposition  of  any  masses  of  iron  in  the 
direct  path  of  the  magnetic  lines. 

It  is  assumed  that  the  reader  has  a  fair  working 
knowledge  of  the  more  important  properties  of  the 
magnetic  circuit ;  he  should  understand,  for  instance,  the 
manner  in  which  the  total  magnetic  flux  in  the  armature 
of  a  dynamo  can  be  approximately  predetermined,  for 
without  a  clear  conception  of  the  fundamental  laws  of  the 
magnetic  circuit,  including  the  peculiar  property  of  iron 

28 


MAGNETIC  FIELD  DUE  TO  CONTINUOUS  CURRENT     29 

(and  one  or  two  other  metals)  of  increasing  the  magnetic 
flux,  the  following  arguments  may  not  be  readily  under- 
stood. 

Apart  from  the  PR  losses  in  conductors,  no  work  has 
to  be  done  in  order  to  maintain  a  magnetic  field ;  but 
energy  was  spent  in  creating  it,  and  this  energy  will  all  be 
given  back  again  to  the  exciting  circuit  when  the  magnetic 
field  is  annulled  or  withdrawn. 

In  order  to  form  a  mental  picture  of  this  property  of  an 
electric  circuit,  consider  a  flywheel,  and  neglect  entirely 
all  questions  of  bearing  friction  or  windage — which,  in 
our  analogy,  are  equivalent  to  the  I2  R  losses  referred  to 
above.  Such  a  flywheel,  once  it  has  attained  a  definite 
speed  of  rotation,  will  continue  to  revolve  for  any  length 
of  time  without  requiring  the  further  application  of  force. 
But  a  force  had  to  be  applied  to  bring  it  up  to  speed,  and 
exactly  the  same  amount  of  energy  as  was  put  into  it 
is  now  available  for  doing  work,  and  will  be  given  back 
again  by  the  time  the  flywheel  has  been  brought  to  rest. 

If  we  multiply  the  total  flux  in  the  core  of  a  dynamo 
field- magnet  by  the  number  of  turns  in  the  exciting 
coil,  we  obtain  a  quantity  which  is  sometimes  referred  to 
as  the  electromagnetic  momentum  of  the  circuit.  There  is 
energy  stored  up  in  such  a  circuit,  and  if  we  attempt 
to  open  it  suddenly,  the  results  are  frequently  disastrous, 
because  this  energy  must  be  transferred  or  dissipated  in 
one  form  or  another,  and  if  we  do  not  provide  a  short- 
circuiting  resistance,  or  some  means  of  drawing  out 
the  arc  slowly,  the  dying  down  of  the  magnetism  will 
induce  such  a  high  E.M.F.  in  the  field  coils  as  to 
break  down  the  insulation  or  form  a  destructive  arc 
at  the  point  of  disconnection.  If,  on  the  other  hand,  we 
short-circuit  the  winding,  or  connect  its  terminals — at  the 
moment  of  switching  off  the  supply — through  an  external 


30  SELF-INDUCTION   AND   CAPACITY 

resistance,  the  current  in  the  coil,  instead  of  being  almost 
instantly  interrupted,  will  die  down  in  the  manner  shown 
on  the  diagram  Fig.  12. 

Here  the  horizontal  distances  from  left  to  right  repre- 
sent lapse  of  time,  and  the  vertical  distances  represent 
current.  It  will  be  noted  that  the  current  falls  rapidly 
at  first,  but  its  rate  of  decrease  diminishes  as  time  goes 
on.  Theoretically,  the  current  never  quite  reaches  zero 


t  Time 

FIG.  12. 

value ;  but,  in  practice,  it  approximates  to  zero  value  in 
a  very  short  space  of  time.  In  most  cases,  the  whole 
distance  from  t  to  t1  would  stand  for  only  a  small  fraction 
of  a  second ;  but  in  the  case  of  a  circuit  with  large  self- 
induction  such  as  the  field  coil  of  a  dynamo,  this  time 
may  extend  over  several  seconds.  The  writer  recollects 
receiving  a  shock  from  a  large  direct-driven  two-pole 
dynamo,  after  the  same  had  been  slowed  down  and  come  to 
a  standstill,  at  the  moment  of  lifting  the  brushes  off  the 


MAGNETIC  FIELD  DUE  TO  ALTERNATING  CURRENT    3 1 

commutator.  This  was  entirely  due  to  the  current  in 
the  field  coils — which  were  short-circuited  through  the 
armature — not  having  quite  died  down,  and  when  the 
circuit  was  opened,  the  energy  still  stored  in  it  took 
the  form  of  a  small  current  at  a  high  potential.  It  should 
not  be  necessary  to  point  out  that,  when  the  current  dies 
down  in  the  manner  indicated  in  Fig.  12,  the  total  energy 
put  into  the  circuit  when  building  up  the  field  is  given 
back  in  the  form  of  I2  R  losses  continuing  for  an 
appreciable  time  in  the  coil  itself,  and  the  external  re- 
sistance (if  any)  which  is  connected  across  the  terminals. 
The  idea  of  the  magnetic  condition  always  existing 
simultaneously  with  the  electric  current  must  not  be  lost 
sight  of.  Thus,  in  Fig.  12,  the  reason  why  the  current 
does  not  drop  instantly  to  zero  is  briefly  this :  if  it  were 
to  do  so,  the  magnetic  flux  in  the  core  (apart  from  the 
residual  magnetism)  would  of  necessity  have  to  drop  like- 
wise instantaneously  ;  but  as,  in  so  doing,  it  would  induce 
a  back  E.M.F.  of  infinite  value,  and  such  as  would  tend 
to  produce  a  current  opposing  the  withdrawal  of  the 
magnetism  ;  it  follows  that  the  current  (and  magnetism) 
in  a  circuit  of  appreciable  resistance  will  actually  die 
down  in  the  manner  indicated  in  Fig.  12 ;  every  decrease 
in  the  current  (and  magnetism)  producing  a  certain 
E.M.F.  of  self-induction  in  the  coil,  tending  to  oppose 
a  more  rapid  rate  of  decrease. 

14.  Magnetic  Field  Due  to  Alternating  Current. 

— When  an  alternating  current  flows  in  a  circuit,  the 
magnetic  condition  of  the  surrounding  medium  will  vary 
in  amount  and  direction  in  accordance  with  the  variations 
of  the  current.  For  any  given  circuit,  the  amount  of  the 
magnetic  flux  may  be  approximately  predetermined  for 
any  instantaneous  value  of  the  current,  provided  we  know 
the  length  and  cross  section,  or  the  magnetic  resistance  of 


32  SELF-INDUCTION    AND   CAPACITY 

the  various  parts  of  the  magnetic  circuit,  and  the  number 
of  turns  in  the  electric  circuit.*  If  there  is  no  iron  or 
other  magnetic  material  in  the  magnetic  circuit,  the  rise 
and  fall  of  the  magnetism  will  synchronise  with  the  rise 


M 


FIG.  13. 


and  fall  of  the  current,  and  not  only  its  maximum  value, 
but  also  every  intermediate  value  will  be  exactly  propor- 

*  If  there  is  iron  in  the  magnetic  circuit,  the  effects  of  hysteresis 
must  be  taken  into  account.  It  should  not  be  necessary  to  remind 
the  reader  that,  when  carrying  a  piece  of  iron  through  a  complete 
cycle  of  magnetisation,  the  ampere-turns  corresponding  to  a  definite 
value  of  the  induction  on  the  rising  portion  of  the  cycle  are  not 
equal  to  those  corresponding  to  the  same  value  of  the  induction  on 
the  descending  portion — i.e.,  when  the  current,  after  having  passed 
through  its  maximum  value,  is  being  reduced. 


FIELD    DUE   TO   ALTERNATING   CURRENT         33 

tional  to  the  strength  of  the  current:  when  the  current 
reaches  zero  value,  the  magnetism  will  also  pass  through 
zero  value ;  and  when  the  current  reverses  in  direction, 
the  magnetic  flux  will  reverse  likewise. 

Thus,  if  we  plot  the  magnetism  corresponding  to  every 
value  of  the  current  in  a  given  circuit  containing  no  iron 


M 


FIG.  14. 

(or  other  magnetic  metal),  we  obtain  a  curve  such  as  that 
shown  in  Fig.  13,  where  I  is  the  current  wave  and  M  the 
magnetism.  Every  ordinate  of  this  latter  curve  is  equal 
to  the  corresponding  ordinate  of  the  curve  I  multiplied 
by  a  constant,  and  the  magnetism  is,  of  course,  exactly 
in  phase  with  the  current. 

In  Fig.  14  the  curve  M  is  such  as  would  be  obtained 

3 


34  SELF-INDUCTION   AND  CAPACITY 

if  the  magnetic  circuit  contained  a  large  amount  of  iron. 
Here,  with  the  one  exception  that  the  maximum  value  of 
the  induction  coincides  with  the  maximum  value  of  the 
current,  it  will  be  noticed  that,  on  the  whole,  the  magnetism 
lags  behind  the  current.* 

A  full  discussion  of  the  various  effects  of  hysteresis,  or 
even  a  detailed  description  of  the  manner  in  which  the 
curve  M  in  Fig.  14  is  derived  from  the  curve  I,  hardly 
enters  into  the  scope  of  the  present  book ;  but  the  reader 
requires  only  an  elementary  knowledge  of  magnetic 
phenomena — especially  as  regards  the  effects  of  iron  in 
a  magnetic  field — to  understand  that  the  residual  magnetism, 
which  requires  a  reversed  magnetising  force  to  eliminate 
it,  is  responsible  for  the  peculiar  relation  of  the  two  curves 
under  consideration. 

15.  E.M.F.  produced  by  an  Alternating  Magnetic 
Field. — We  know  that  the  E.M.F.  generated  in  a  con- 
ductor passing  through  a  magnetic  field  is  directly  propor- 
tional to  the  rate  at  which  the  conductor  is  cutting  the 
(imaginary)  magnetic  lines — or  tubes — of  induction.  If 
100,000,000  maxwells  or  C.G.S.  magnetic  lines  are  cut 
during  one  second  of  time,  the  average  value  of  the 
resulting  E.M.F.  will  be  i  volt.  If  we  thrust  a  magnet 
into  a  coil  of  wire,  the  E.M.F.  generated  in  the  coil  is 
proportional  to  the  strength  of  the  magnet  and  the  rapidity 
with  which  it  has  been  thrust  into  the  coil.  The  direction 
of  this  E.M.F.  is  always  such  as  to  tend  to  produce  a 
current  which  will  oppose  the  alteration  in  the  magnetic 
condition. f  If ,  on  inserting  the  magnet,  a  positive  E.M.F. 

•~:  Hence  the  word  hysteresis  originally  suggested  by  Professor 
Ewing. 

f  There  is  no  exception  to  this  rule,  which  is  known  as  Lenz's 
law. 


E.M.F.   DUE   TO   ALTERNATING    MAGNETISM      35 

was  generated,  a  negative  E.M.F.  will  be  induced  when 
the  magnet  is  withdrawn. 

In  Fig.  15,  let  the  curve  M  represent  the  rise  and  fall 
of  the  alternating  magnetism  due  to  an  alternating  electric 
current — which  has  not  been  drawn  in  the  diagram,  but 
which,  if  there  is  no  iron  in  the  circuit,  will  be  in  phase 
with  M,  and,  if  there  is  iron  in  the  circuit,  will  be  in 


FIG.  15. 


advance  of  M  (see  Figs.  13  and  14).  Let  us  see  what  s 
the  E.M.F.  which  this  varying  magnetism  will  induce  in 
an  electric  circuit  with  which  it  is  linked.  In  the  first 
place,  it  is  evident  that,  if  the  magnetism  were  of  a 
constant  value — however  large — no  E.M.F.  would  be 
generated  in  the  circuit.  In  other  words,  if  its  rate  of 
change  were  nil,  the  value  of  the  induced  E.M.F.  would 
be  zero.  This  is  exactly  what  occurs  at  the  point  A, 
when  the  curve  M  has  reached  its  maximum  value  ;  the 
amount  of  magnetism  is  neither  increasing  nor  decreasing 


36  SELF-INDUCTION   AND   CAPACITY 

at  this  particular  instant,  and  consequently  there  can  be 
no  E.M.F.  generated.  We  can,  therefore,  plot  the  point 
a  of  the  E.M.F.  curve  on  the  horizontal  datum  line 
immediately  below  the  maximum  value  of  the  magnetism 
curve — this  maximum  value  being  graphically  defined  by 
the  point  of  contact  of  the  horizontal  tangent  to  the  curve 
M,  as  indicated  by  the  dotted  line. 

Consider,  now,  the  point  b  on  the  curve  M.  This  point 
represents  not  only  the  instant  of  time  corresponding  to 
the  reversal  in  direction  of  the  magnetism ;  but,  in  this 
example,  it  also  indicates  that  portion  of  the  cycle  where 
the  rate  of  change  in  the  magnetism  is  greatest — as  evi- 
denced by  the  slope  of  the  curve  being  steepest  at  this 
point.  If  this  change  is  at  the  rate  of  100,000,000  C.G.S. 
magnetic  lines  withdrawn  from  the  circuit  per  second,  an 
E.M.F.  of  i  volt  will  be  generated  in  every  turn  of  the 
electric  circuit  which  is  linked  with  this  magnetism. 

Thus,  if  we  know  the  number  of  turns  in  the  circuit 
and  also  the  scale  to  which  the  curve  M  has  been  drawn, 
we  can  readily  calculate  the  actual  induced  E.M.F.  at 
the  instant  b  and  plot  this  value  b  B  to  any  convenient 
scale. 

As  to  the  sign  of  this  E.M.F.,  it  will,  at  this  point,  be 
positive  (and  therefore  must  be  plotted  above  the  datum 
line)  because,  since  the  magnetism  is  falling  from  a 
positive  maximum  towards  a  negative  maximum,  it  is  a 
positive  E.M.F.  which  is  necessary  to  produce  a  current 
such  as  would  oppose  or  counteract  this  decrease  in  the 
magnetism. 

We  have  seen  that,  where  the  curve  M  is  horizontal, 
the  induced  volts  are  nil;  if  it  were  possible  for  this 
curve  to  drop  so  rapidly  as  to  become  perpendicular  to 
the  horizontal  datum  line,  this  would  indicate  an  instan- 
taneous change  in  the  magnetic  condition,  or  an  infinitely 


E.M.F.   DUE   TO   ALTERNATING   MAGNETISM      37 

great  rate  of  change,  and,  therefore,  an  infinitely  great 
induced  E.M.F.  whatever  might  be  the  actual  number  of 
magnetic  lines  added  to  or  withdrawn  from  the  circuit  ; 
and  if  the  reader  has  a  clear  understanding  of  these 
extreme  conditions  he  will  probably  accept  the  statement 
that  the  steepness  or  slope  of  the  curve  M  is,  at  every  point, 
an  exact  measure  of  the  rate  of  change  in  the  magnetic 
condition.* 

As  an  example,  suppose  the  ordinate  p  P  represents 
100,000  C.G.S.  lines  of  magnetism.  Draw  the  tangent 
O  P  to  the  curve  M  at  the  point  P,  and  measure  O  p  :  let 
us  assume  that  this  distance  corresponds  to  the  two- 
hundredth  part  of  a  second.  The  increase  in  the 
magnetism  at  the  point  P  is,  therefore,  at  the  rate  of 
100,000  C.G.S.  lines  in  the  two-hundredth  part  of  a 
second,  or  twenty  million  lines  per  second  ;  and  if  we 
assume  that  there  are  one  hundred  turns  of  wire  in  the 
circuit,  the  instantaneous  value  of  the  induced  E.M.F.  at 
the  moment  p  in  the  hundred  turns  of  wire  will  be 

20.000,000  xioo 


100,000,000 

and  this  value,  represented  by  p  e,  must  be  plotted  below 
the  datum  line,  because  this  E.M.F.  will  be  such  as  will 
tend  to  produce  a  current  in  a  negative  direction  —  i.e.,  such 
as  would  oppose  the  variation  in  the  magnetic  flux,  which, 
at  this  moment,  is  increasing  in  amount. 

The  circuit  we  have  been  considering  might  be  the 
armature  winding  of  an  alternator  or  the  secondary  coils 

*  It  would  be  an  easy  matter  to  prove  this  statement,  and  in  all 
probability  the  reader  will  be  quite  capable  of  doing  this  to  his 
own  satisfaction.  The  author's  purpose  is  not  to  prove  every 
statement  made  in  the  course  of  this  and  subsequent  chapters  ;  but 
a  general  explanation,  suggestive  of  the  lines  on  which  more  exact 
proofs  may  be  sought,  will,  wherever  possible,  be  given. 


38  SELF-INDUCTION    AND   CAPACITY 

of  a  transformer,  and,  in  either  case,  the  induced  E.M.F., 
as  shown  in  Fig.  15,  would  be  a  quarter  period — or 
90  degrees — in  advance  of  the  alternating  magnetism  to 
which  it  owes  its  existence  ;  and,  moreover,  since  exactly 
the  same  total  magnetic  flux  as  is  threaded  through  the 
coils  during  one  half-period  is  withdrawn  during  the  suc- 
ceeding half-period,  the  mean  value  of  the  positive  half- 
wave  of  the  induced  E.M.F.  is  always  exactly  equal  to 
the  mean  value  of  the  negative  half-wave.* 

If  N  is  the  total  amount  of  magnetic  flux  produced  by 
the  maximum  value  of  the  alternating  current  passing 
through  the  coil,  and  if  S  stands  for  the  number  of  turns 
in  the  coil  and  /  for  the  frequency,  then  the  mean  value  of 
the  induced  E.M.F.  is  proportional  to  N  S  /,  and  this 
relation  holds  good  whatever  may  be  the  shape  of  the 
current  wave  producing  the  magnetism.  (See  Appendix  I. 
at  end  of  book.) 

It  must  not  be  supposed  that  because  the  mean  value 
of  the  E.M.F.  is  independent  of  the  wave  form,  the 
/x/mean  square  value  is  likewise  unaffected  by  the  shape 
of  the  wave :  on  the  contrary,  the  //mean  square  value 
bears  no  definite  relation  to  the  mean  value,  but  depends 
largely  upon  the  wave  form,  which  it  is  very  difficult  to 
predetermine  with  accuracy. 

1 6.  Inductance. — The  changes  in  the  magnetic  flux 
of  induction  linked  with  a  given  electric  circuit,  as  repre- 
sented by  the  curve  M  of  Fig.  15  in  the  preceding  article, 
may  conceivably  be  due  to  changes  in  the  current  carried 
by  this  particular  circuit.  In  such  case,  the  curve  El 
would  represent  the  counter  E.M.F.  of  self-induction.  The 
self-induction  of  a  circuit,  as  indeed  is  indicated  by  the 

*  The  time  integrals  of  the  positive  and  negative  half-waves  are 
equal,  whatever  may  be  the  law  of  variation  of  the  alternating 
magnetism. 


INDUCTANCE  39 

expression  itself,  is  the  induction  or  total  number  of 
magnetic  lines  due  to  the  current  flowing  in  the  circuit.  For 
any  particular  current  I  in  the  conductor  forming  the 
electric  circuit,  the  total  magnetic  flux  of  induction  linked 
with  this  circuit  which  is  due  to  the  current  1  is  the  self- 
induction  corresponding  to  that  particular  current. 

What  is  known  as  the  coefficient  of  self -induction,  or,  more 
simply,  the  inductance  of  a  circuit,  is  a  multiplier — generally 
denoted  by  the  letter  L — which  takes  into  account,  not 
only  the  amount  of  the  total  flux  of  induction  due  to  unit 
current  flowing  in  the  circuit,  but  also  the  number  of 
times  the  induction  is  linked  with  the  electric  circuit. 
Thus  the  inductance,  L,  might  be  denned  as  the  amount 
of  self-enclosing  of  magnetic  lines  by  the  circuit  when  the 
current  has  unit  value.  It  cannot  be  expressed  in  maxwells, 
since  it  is  equal  to  maxwells  x  number  of  turns.  The  prac- 
tical coefficient  of  self-induction  is  the  henry.  If  the 
number  of  maxwells  representing  the  flux  N  is  known 
when  the  maximum  value  of  the  current  is  I  amperes, 
and  if  the  circuit  carrying  the  current  I  is  linked  S  times 
with  the  flux  N,  then  the  inductance,  in  henrys,  is 


This  coefficient  is  constant  only  for  a  circuit  containing 
no  iron.  The  presence  of  iron  leads  to  the  flux  N  being 
no  longer  proportional  to  the  current  I,  and  L  will  there- 
fore be  a  function  of  I. 

It  is  not  proposed  to  make  much  use  of  the  quantity  L 
in  this  book ;  but  it  is  convenient,  and  of  frequent  occur- 
rence, in  the  analytical  treatment  of  alternating-current 
problems. 

17.  Reactance.— In  a  circuit  carrying  a  continuous 
current,  the  E.M.F.  can  be  expressed  in  terms  of  the 


40  SELF-INDUCTION    AND   CAPACITY 

current  and  resistance  by  writing  Ohm's  law  in  the  form 
E=IxR.  We  have  just  seen  how  there  is  another 
E.M.F.,  called  the  E.M.F.  of  self-induction,  which  very 
frequently  asserts  its  presence  in  a  circuit  carrying  an 
alternating  current,  and  the  term  reactance  has,  for  con- 
venience, been  given  to  the  ratio  obtained  by  dividing 
the  E.M.F.  of  self-induction  by  the  current  in  the  circuit. 
Thus  we  may  write. 

E.M.F.  of  self-induction  =  current  x  reactance, 

the  last  term  being  a  multiplier  which  can  be  expressed 
in  ohms,  and  which  will  be  directly  proportional  to  the 
inductance  of  the  circuit  and  to  the  frequency  of  the 
current,  but  which  will  also  depend,  to  a  certain  extent, 
upon  the  wave  form  of  the  alternating  current. 

Thus,  if  a  circuit  has  large  inductive  reactance,  the 
induced  E.M.F.,  for  a  given  current,  will  be  greater 
than  if  the  reactance  is  small.  A  load  of  incandescent 
lamps  is  an  example  of  a  circuit  of  small  reactance, 
whereas  the  primary  winding  of  a  transformer — which, 
in  addition  to  enclosing  a  large  magnetic  flux,  has  also  a 
considerable  number  of  turns — is  a  circuit  of  large  react- 
ance ;  but  this  expression  has  no  meaning  except  in 
relation  to  a  circuit  carrying  an  alternating  current  at 
a  definite  frequency.  In  fact,  the  chief  reason  why  refer- 
ence has  been  made  to  this  term  is  that  it  is  used  by 
most  writers  when  treating  of  alternating  currents,  and 
without  an  elementary  knowledge  of  the  meaning  of  such 
terms,  the  reader  might  be  seriously  handicapped  when 
taking  up  more  advanced  works  on  the  subject. 

If  it  is  permissible  to  assume  the  sine  wave  variation 
of  E.M.F.  and  current,  it  can  be  shown  that  the  inductive 
reactance  is 

X    =    2   7T/L, 


CIRCUIT  OF   NEGLIGIBLE   INDUCTANCE          41 

where  L  is  the  coefficient  of  self-induction  of  the  circuit, 
expressed  in  henrys.  The  counter  E.M.F^of  self-induc- 
tion is,  therefore, 

X    I    =    27T/L   I, 

where  I  is  the  R.M.S.  value  of  the  current,  in  amperes, 
if  it  is  the  corresponding  value  of  the  E.M.F.  that  it  is 
required  to  calculate. 

1 8.  Current  Flow  in  Circuit  of  Negligible  Induc- 
tance.— Let  us  consider  an  electric  circuit  which  is 
practically  without  self-induction,  or  electrostatic  capacity. 
It  may  consist  of  a  wire  doubled  back  upon  itself  (in  the 
manner  adopted  in  winding  resistance  coils  for  testing 
purposes),  or  of  glow  lamps,  or  of  a  water  resistance. 

If  an  alternating  E.M.F.  is  applied  to  the  terminals  of 
such  a  circuit,  the  current  at  any  instant  will  be  equal  to 
the  ratio  of  the  instantaneous  value  of  the  E.M.F.  to  the 
total  resistance  of  the  circuit ;  or, 

T         -   ^inst- 

R     ' 

from  which  we  see  that  the  current  wave  will  be  of  the 
same  shape  as  the  E.M.F.  wave,  and  in  phase  with  it — a 
state  of  things  which  is  the  evident  result  of  the  fulfilment 
of  Ohm's  law  :  for  there  is  no  reason  for  supposing  that 
Ohm's  law  is  not  equally  applicable  to  variable  as  to 
steady  currents  ;  it  is  only  necessary  to  bear  in  mind  that, 
in  the  case  of  variable  currents,  the  applied  E.M.F.  and 
the  resultant  E.M.F.  in  the  circuit  (to  which  the  current 
is  due)  are  not  necessarily  one  and  the  same  thing.  In 
the  case  under  consideration,  of  a  circuit  supposed  to  be 
without  self-induction  or  capacity,*  there  is  only  one 
E.M.F.  tending  to  produce  a  flow  of  current— i.e.,  the 

*  The  effects  of  capacity  will  be  dealt  with  in  due  course. 


42  SELF-INDUCTION    AND   CAPACITY 

E.M.F.  supplied  at  the  terminals  of  the  generator:  the 
current  will  therefore  rise  and  fall  in  exact  synchronism 
with  the  applied  E.M.F. 

19.  Current  Flow  in  Circuit  of  Appreciable  In- 
ductance.— In  order  to  get  a  better  understanding  of 
the  whole  question  of  self-induction  in  connection  with 
alternating  currents,  let  us  consider  an  alternating  current 
flowing  in  a  circuit  which  has  both  ohmic  resistance  and 
reactance — as,  for  instance,  a  coil  of  wire  of  many  turns, 
which,  for  the  present,  we  will  assume,  has  no  iron  core. 
Such  a  current  is  shown  graphically  by  the  curve  I  in 
Fig.  1 6,  where  intervals  of  time  are  measured,  as  usual, 
horizontally  from  left  to  right.  The  magnetism  due  to 
the  current  I  will  vary  in  amount  and  direction  in  ac- 
cordance with  the  variations  of  the  current.  It  may  be 
calculated  in  the  usual  way  for  any  given  value  of  I, 
provided  we  know  the  length  and  cross-section,  or  the 
magnetic  resistance  of  the  various  parts  of  the  magnetic 
circuit,  and  the  number  of  turns  of  wire  in  the  coil.  Let 
the  curve  m  represent  the  rise  and  fall  of  this  magnetism. 

Since  the  induced  or  counter  E.M.F.  due  to  these  varia- 
tions in  the  magnetic  induction  will  be  proportional  to 
the  rate  of  change  in  the  total  number  of  magnetic  lines 
threaded  through  the  circuit,  we  shall  have  no  difficulty 
in  drawing  the  curve  Ex  (as  in  article  15,  p.  34)  to  re- 
present the  E.M.F.  of  self-induction,  which  lags  exactly 
one-quarter  of  a  period  behind  the  current  wave. 

We  are  now  in  a  position  to  determine  the  potential 
difference  which  must  exist  at  the  terminals  of  the 
circuit  in  question,  in  order  that  the  current  I  will  flow 
through  it. 

Draw  the  curve  E2  to  represent  the  E.M.F.  required 
to  overcome  the  ohmic  resistance.  It  will  be  in  phase 
with  the  current,  because  its  value  at  any  point  is  simply 


CIRCUIT   OF  APPRECIABLE   INDUCTANCE         43 

I  X  R,  where  R  stands  for  the  resistance  of  the  circuit. 
Now  add  the  ordinates  of  E2  to  those  of  an  imaginary 
curve  exactly  similar  but  opposite  to  Elt  and  the  resulting 
curve  E  will  evidently  be  that  of  the  impressed  potential 
difference  which,  if  maintained  at  the  ends  of  the  circuit 
under  consideration,  will  cause  the  current  I  to  flow  in  it. 
Thus  we  see  how  the  relation  between  the  impressed 
E.M.F.  and  the  resulting  current  may  be  graphically 
worked  out  for  any  given  case. 


\ 


FIG.  16. 

From  a  study  of  the  curves  in  Fig.  1 6  it  is  evident  that 
the  effect  of  self-induction  is  to  make  the  current  lag 
behind  the  impressed  E.M.F.  If  the  E.M.F.  required 
to  force  the  current  against  the  ohmic  resistance  is  small 
in  comparison  with  the  induced  E.M.F.,  the  lag  will  be 
very  considerable ;  it  cannot,  however,  exceed  one- 
quarter  of  a  complete  period,  which  limit  is  only  reached 
when  the  E.M.F.  of  self-induction  is  so  large,  and  the 
ohmic  resistance  of  the  circuit  so  small,  as  to  render 


44  SELF-INDUCTION    AND   CAPACITY 

the  E.M.F.  required  to  overcome  this  resistance  of  no 
account. 

In  order  briefly  to  sum  up  the  principles  governing  the 
flow  of  an  alternating  current  in  an  inductive  circuit,  we 
may  say  that  the  varying  current  produces  changes  of 
magnetism,  which  again  produce  a  varying  E.M.F.,  called 
the  "  E.M.F.  of  self-induction."  This,  together  with  the 
E.M.F.  already  existing  (and  without  which  no  current 
would  flow),  produces  the  useful  or  resultant  E.M.F. 
By  dividing  the  value  of  this  resultant  E.M.F.  at  any 
instant  by  the  total  ohmic  resistance  of  the  circuit,  the 
corresponding  current  intensity  is  obtained.  This  con- 
dition must  always  be  fulfilled,  otherwise  Ohm's  law  would 
not  be  satisfied. 

20.  Diagram  showing  Relation  of  E.M.F.S  in 
Inductive  Circuit. — We  have  seen  in  article  15,  p.  34, 
and  again  in  discussing  the  curves  of  Fig.  16,  that  the 
induced  E.M.F.  lags  exactly  90  degrees  behind  the 
magnetism ;  and  since  this  magnetism  rises  and  falls  in 
synchronism  with  the  current  to  which  it  owes  its  origin, 
the  same  relation — i.e.,  a  phase  difference  of  a  quarter 
period — exists  between  the  induced  E.M.F.  and  current 
in  an  inductive  circuit,  such  as  we  have  been  considering. 

Again,  the  resultant  E.M.F.  required  to  overcome  the 
ohmic  resistance  of  the  circuit  is  necessarily  in  phase 
with  the  current  (refer  to  the  curve  E2  in  Fig.  16),  and 
both  these  relations  hold  good,  whatever  may  be  the 
shape  of  the  current  wave,  provided  always  that  there 
is  no  iron  or  other  magnetic  metal  in  the  circuit.  If, 
therefore,  we  wish  to  draw  the  vector  diagram  of  the 
E.M.F.s  in  such  an  inductive  circuit,  the  two  vectors 
O  E!  and  O  E2  (Fig.  17),*  representing  respectively  the 

*  Compare  with  Fig.  n,  p.  24. 


RELATION   OF   E.M.F.S   IN   INDUCTIVE   CIRCUIT      45 

induced  and  the  useful  or  energy  E.M.F.s,  must  be  drawn 
at  right  angles  to  each  other,  with  O  E2  in  advance  of  O  Et. 
The  necessary  impressed  E.M.F.  at  the  terminals  of 
the  circuit  is  found — as  explained  in  article  19 — by  com- 
pounding the  E.M.F.  E2  with  an  imaginary  E.M.F.  (O^) 
exactly  equal,  but  opposite  in  phase  to  Er  In  this 
manner  the  vector  O  E  is  obtained. 


FIG.  17. 


Now  complete  the  triangle  E  O  E2,  and  note  that  the 
lengths  of  the  sides  of  such  a  triangle  represent  re- 
spectively the  resultant  or  useful  E.M.F.;  the  counter 
E.M.F.  or  E.M.F.  of  self-induction;  and  the  total  im- 
pressed E.M.F.  at  the  terminals  of  the  circuit.  Moreover, 
since  the  side  O  E  is  the  hypotenuse  of  a  right-angled 
triangle,  this  latter  quantity  (the  impressed  E.M.F.)  is 
always  equal  to  the  square  root  of  the  sum  of  the  squares 


46 


SELF-INDUCTION   AND   CAPACITY 


of  the  other  two,  which  is  an  easy  and  useful  rule  to  bear 
in  mind. 

It  should  not  be  necessary  to  remind  the  reader  that 
the  angle  9  is  the  angle  of  lag  between  the  virtual 
and  the  impressed  volts,  and  that  cos  0,  or  the  ratio 
O  E2  ^-  O  E  is  the  power  factor  of  this  particular  circuit. 

In  calculating  the  various  E.M.F.'s  in  the  circuit,  as 
drawn  in  Fig.  17,  the  value  of  the  current  has  neces- 
sarily been  taken  into  account.  Let  us  eliminate  this 
factor  entirely,  by  dividing  the  three  quantities  O  Elf 
O  E2,  and  E  E2  by  the  value  of  the  current.  The 


O  Resistance  C2 

FIG.  1 8. 

triangle  need  not  be  altered  in  any  way ;  it  has  merely 
to  be  considered  as  having  been  drawn  to  a  different 
scale.  It  is  reproduced  in  Fig.  18,  and  it  will  be  noted 
that  the  line  O  E2  now  stands  for  the  resistance  of  the 
circuit,  while  E  E2  represents  the  reactance  (see  article  17, 
p.  39).  With  regard  to  O  E,  since  this  is  the  result 
obtained  by  dividing  the  total  E.M.F.  by  the  current, 
it  evidently  stands  for  the  apparent  resistance  of  the 
circuit,  to  which  the  name  impedance  has  been  given :  this 
quantity,  and  also  the  reactance,  can  be  expressed  in 
ohms. 


EFFECT   OF   IRON   IN   MAGNETIC   CIRCUIT        47 
We  may,  therefore,  write  : 


amperes 
and  again, 


impedance  =  */  (resistance)2  +  (reactance)2. 

It  must  not  be  overlooked  that  such  expressions  and 
relations  as  these  have  only  a  limited  usefulness,  and  the 
quantity  understood  by  the  word  impedance  is  merely 
a  multiplier,  which  is  not  constant,  even  for  a  given 
circuit,  but  which  depends  upon  the  frequency  and  wave 
form  of  the  impressed  E.M.F. 

A  little  consideration  will  make  it  clear  that  the  higher 
the  periodicity,  the  more  important  becomes  the  reactance 
relatively  to  the  resistance,  and  that,  in  all  cases  where  the 
counter  E.M.F.  is  objectionable,  it  is  advisable  to  keep 
the  periodicity  as  low  as  possible. 

21.  Effect  of  Iron  in  Magnetic  Circuit. — Although 
it  is  not  intended  to  investigate  the  causes  which  lead  to 
a  distortion  of  the  current  wave,  and  a  certain  loss  of 
energy  when  iron  is  introduced  into  the  magnetic  circuit, 
it  will,  nevertheless,  be  necessary  to  note  briefly  the  effects 
produced  when  iron  is  used — as  in  generators,  motors, 
and  transformers — for  the  purpose  of  increasing  the  mag- 
netic induction  and  concentrating  it  at  certain  points. 

There  are  two  causes  of  loss  of  energy  when  the  path 
of  the  magnetic  lines  is  through  iron ;  the  first  is  due  to 
eddy  currents,  and  the  second  to  hysteresis. 

22.  Eddy  Currents. — Whenever  a  conductor  of  elec- 
tricity is  placed  in  a  fluctuating  magnetic  field,  Foucault 
or  eddy  currents  are  produced.     These  eddy  currents  are 
in  phase  with  the  induced  E.M.F.,  and  they  will  tend 
to  oppose  the  changes  in  the  magnetism.     Other  con- 


48  SELF-INDUCTION    AND   CAPACITY 

ditions  being  similar,  the  mean  demagnetising  tendency 
of  such  currents  will  be  directly  proportional  to  the 
specific  conductivity  of  the  metal  in  the  path  of  the 
magnetic  lines ;  the  power  necessary  to  maintain  these 
currents  is  dissipated  in  the  form  of  heat  in  the  mass 
of  the  metal,  and  this  loss  of  power  would  be  very  con- 
siderable unless  special  precautions  were  taken  to  reduce 
it.  The  remedy — which  is  adopted  in  the  cores  of  all 
alternating-current  machinery — is  to  laminate  or  divide 
the  metal  core  in  a  direction  parallel  to  the  magnetic 
flux,  and  slightly  insulate  the  adjacent  plates  or  wires 
from  each  other.  In  this  manner  the  losses  may  be 
reduced  to  a  very  small  amount. 

The  power  dissipated  in  the  laminated  iron  forming 
the  cores  of  transformers  and  alternating  -  current 
machinery  is  proportional  to  the  square  of  the  E.M.F. 
induced  in  the  local  circuits  in  which  eddy  currents 
circulate ;  it  is  therefore  approximately  proportional  to 
the  square  of  the  maximum  value  of  the  induction,  and 
can  be  expressed  by  the  formula 

watts  per  pound  =  — ^  (tf  B)2, 

where  t  —  thickness  of  laminations   in   inches   (usually 

about  -014) ; 

/  =  frequency  in  periods  per  second  ; 
B  =  maximum  value  of  the  induction  in  gausses 
(C.G.S.  lines  per  square  centimetre). 

23.  Hysteresis. — The  losses  due  to  hysteresis  occur 
only  in  the  case  of  the  magnetic  lines  passing  through 
a  magnetic  metal,  such  as  iron,  and  they  are  in  no  wise 
dependent  upon  the  degree  of  lamination  of  the  magnetic 
circuit. 


HYSTERESIS  49 

It  is  well  known  that  all  iron,  even  the  softest  and 
purest,  retains  some  magnetism  after  the  magnetising 
force  has  been  removed.  By  applying  a  magnetising 
force  in  the  opposite  direction  this  residual  magnetism 
is  destroyed,  and  the  magnitude  of  this  force — or,  in 
other  words,  the  amount  of  work  which  has  to  be  done 
to  withdraw  this  magnetism — depends  upon  the  quality  of 
the  iron.  Soft  annealed  wrought  iron  retains  most  mag- 
netism ;  but,  on  the  other  hand,  it  parts  with  it  more 
easily  than  the  harder  qualities  of  iron  and  steel,  and  for 
this  reason  requires  the  least  expenditure  of  energy  to 
carry  it  through  a  given  cycle  of  magnetisation. 

The  frequency,  or  number  of  alternations  of  the  mag- 
netism per  second,  does  not  appreciably  influence  the 
loss  through  hysteresis  per  cycle,  and,  therefore,  the  loss 
of  power  per  cubic  inch  or  per  pound  of  iron  in  the  path 
of  the  magnetic  lines  will  be  proportional  to  the  fre- 
quency. It  has  also  been  ascertained  experimentally 
that  the  energy  expended  in  carrying  a  given  sample  of 
iron  through  one  complete  cycle  of  magnetisation  is 
approximately  proportional  to  the  i'6th  power  of  the 
limiting  induction ;  and  if  B  stands  for  the  maximum 
value  of  the  induction,  we  may,  therefore,  write 

watts  lost  per  pound  of  iron  oc  B1'6  x  frequency. 

In  the  case  of  well-annealed  commercial  iron  stampings, 
the  hysteresis  loss  per  pound  at  a  frequency  of  50  periods 
per  second  would  be  about  '09  watt  with  a  limiting  induc- 
tion of  15,000  C.G.S.  lines  per  square  inch  of  cross-sectional 
area,  and  about  -25  watt  at  double  this  induction — i.e.t 
30,000  C.G.S.  lines. 

Assuming  the  iron  stampings  to  be  -014  in.  thick,  and 
sufficiently  well  insulated  from  each  other,  the  eddy- 
current  losses  for  the  same  frequency  and  maximum 

4 


50  SELF-INDUCTION   AND   CAPACITY 

induction  would  be  approximately  '02  and  '07  watt  per 
pound  ;  from  which  it  will  be  seen  that  the  losses  due  to 
eddy  currents  in  the  iron  cores  are  of  less  importance 
than  the  hysteresis  losses,  and,  indeed,  with  low  induc- 
tions and  low  frequencies  the  eddy-current  losses  are 
almost  negligible. 

Although  the  losses  through  hysteresis  vary  consider- 
ably with  different  qualities  of  iron  or  steel,  the  following 
formula  may  be  useful  for  approximate  calculations.  It 
gives  the  watts  per  pound  for  a  good  quality  of  trans- 
former iron : 

w=113fBi>™ 

io8 

— where  B  is  the  maximum  value  of  the  induction,  in 
gausses  or  C.G.S.  lines  per  square  centimetre.  To  obtain 
watts  lost  per  cubic  inch,  multiply  by  -28. 

In  the  case  of  silicon  steel  sheets,  frequently  used  for 
transformer  cores,  the  loss  might  be  only  75  per  cent,  of 
the  amount  given  by  the  above  formula. 

24.  General  Conclusions  regarding  the  Intro- 
duction of  Iron  in  the  Magnetic  Circuit. — Let  us 

consider  a  choking  coil  consisting  of  a  number  of  turns 
of  wire,  and  draw  two  vector  diagrams  for  such  a  circuit : 
the  first  (Fig.  19)  showing  relation  of  current  to  E.M.F. 
on  the  assumption  that  the  solenoid  or  coil  of  wire  has 
an  air  core — i.e.,  no  iron  core — and  the  second  (Fig.  20) 
showing  what  takes  place  in  the  same  coil  when  an 
iron  core  is  introduced  for  the  purpose  of  reducing  the 
current. 

For  the  sake  of  comparison,  we  shall  assume,  in  the 
first  place,  that  the  resistance,  R,  of  the  coil  is  constant 
in  both  cases,  and  equal  to  4  ohms ;  and  in  the  second 
place  we  shall  see  what  are  the  relations  of  impressed 


INTRODUCTION  OF  IRON  IN  MAGNETIC  CIRCUIT       51 

E.M.F.  and  current  to  induce  in  each  case  a  back  E.M.F. 
of  100  volts.* 

In  Fig.  19  (which  should  be  compared  with  Fig.  u, 
p.  24)  draw  O  Ex  of  such  a  length  as  to  represent  the 
induced  E.M.F.  of  100  volts,  and  O  I  exactly  90  degrees 
in  advance,  to  represent  the  magnetising  current — that  is 
to  say,  the  alternating  current  of  which  the  maximum 
value  will  produce  the  necessary  magnetic  flux  to  induce 
the  E.M.F.  of  100  volts  in  the  windings  of  the  coil.  We 
shall  suppose  this  current,  I,  to  be  exactly  10  amperes. 
The  E.M.F.,  E2,  required  to  send  this  current  against 
the  resistance  of  4  ohms  will  be 

E2  =  10  x  4  =  40  volts, 

and  this  component  of  the  total  E.M.F.  must  be  drawn  in 
phase  with  O  I.  The  total  necessary  impressed  E.M.F.,  E, 
is  obtained  by  compounding  O  E2  and  an  imaginary 
E.M.F.  exactly  equal,  but  opposite  to  O  Er  This  will 
give  us  the  value  of  E  as  about  108  volts,  and  the  angle 
of  lag  0  =  68  degrees. 

It  will  be  noticed  that  the  current  lags  considerably 
behind  the  impressed  E.M.F.,  and  this  lag  would  approach 
more  and  more  nearly  to  its  limiting  value  of  90  degrees 
if  the  resistance  of  the  coil  were  diminished  :  if  it  were 
possible  to  make  R  =  o,  the  current  I  would  be  actually 
wattless  ;  as  it  is,  the  power  lost  in  the  choking  coil  is  all 
spent  in  heating  the  wire — the  alternating  magnetic  flux 

*  The  relation  between  magnetic  flux  and  current  was  discussed 
in  article  14  ;  and  in  article  15  it  was  explained  how  the  induced 
E.M.F^  can  be  calculated  for  any  particular  point  in  the  cycle  of 
magnetisation.  We  shall,  therefore,  assume  that  the  current 
required,  in  the  present  example,  to  produce  the  given  induced 
E.M.F.  of  100  volts,  can  be  readily  calculated  ;  but  if  the  reader 
wishes  to  go  more  fully  into  the  question,  he  is  referred  to  Appen- 
dix I.  at  the  end  of  the  book. 


52  SELF-INDUCTION    AND   CAPACITY 

costs  nothing  to  produce.  The  actual  power  lost  in  our 
example  is  400  watts.  It  is  equal  to  I2  R  or  to  O  I  x  O  E2, 
or  to  O  I  x  O  E  x  cos  0. 

We  shall  now  suppose  that  the  magnetic  circuit,  instead 
of  being  entirely  through  air,  is  made  up  either  partly  or 


FIG.  19. 

wholly  of  iron,  which  must  be  laminated  to  prevent  an 
abnormal  loss  through  eddy  currents. 

Draw  O  Ex  in  Fig.  20,  as  before,  to  represent  100  volts ; 
but  O  Im — i.e.,  the  necessary  magnetising  current — will 
not  be  so  great  as  O  I  in  Fig.  19,  because  the  introduc- 


INTRODUCTION  OF  IRON  IN  MAGNETIC  CIRCUIT      53 

tion  of  the  iron  core  has  provided  an  easier  path  for  the 
magnetism.  We  shall  suppose  that  O  Im  =  5  amperes, 
or  half  the  magnetising  current  of  the  previous  example. 
This  is  not,  now,  the  total  current  in  the  coil,  but  merely 


FIG.  20. 


that  component  of  the  total  current  which  is  required  to 
produce  the  necessary  magnetic  flux :  it  must  be  drawn, 
as  before,  exactly  90  degrees  in  advance  of  O  Ej. 

Let  us  suppose  that  the  eddy  currents  in  the  iron  are 


54  SELF-INDUCTION    AND   CAPACITY 

responsible  for  a  loss  of  power  equal  to  150  watts,  and 
that  the  hysteresis  is  responsible  for  300  watts.  This  total 
power  of  450  watts  has  to  be  supplied  by  the  generator — 
or  whatever  source  of  supply  the  choking  coil  is  connected 
to— in  the  form  of  a  component  of  the  total  current 
flowing  against  the  induced  E.M.F.,  the  portion  of  the 
current  required  to  provide  for  the  eddy-current  loss 

being  -2—  =  i  "5  amperes,  while  the  hysteresis  component 
100 

of  the  current  will  be  ^      =  3  amperes. 
100 

These  two  components  of  the  total  current  are  shown 
as  O  Ie  and  O  Ih  in  Fig.  20,  and  they  are  drawn  in  a 
direction  exactly  opposite  to  O  Ex  (the  induced  E.M.F.). 
The  total  current  O  I  is  obtained  by  compounding  O  Im 
and  O  lw,  the  last  vector  being  equal  to  the  sum  of  the 
two  energy  components,  O  Ie  and  O  Ih. 

As  regards  the  E.M.F.,  the  volts  required  to  overcome 
the  ohmic  resistance  of  the  coil  are,  as  before,  in  phase 
with  the  total  current  O  I ;  and  since  this  measures  very 
nearly  7  amperes,  the  vector  O  E2  must  be  drawn  of  such 
a  length  as  to  represent  7  x  4  =  28  volts. 

The  resultant  or  total  necessary  impressed  E.M.F. 
is  obtained  by  compounding  O  E2  and  the  imaginary 
E.M.F.  O  E'-L  exactly  equal  and  opposite  to  O  Er  It 
will  be  found  that  the  angle  of  lag,  0,  is  now  about 
38  degrees,  while  the  total  power  lost  =  O  I  x  O  E  cos  6  = 
about  650  watts. 

Thus,  although  the  I2  R  losses  are  less,  owing  to  the 
reduction  of  current  due  to  the  introduction  of  an  iron 
core,  the  losses  in  this  core  are  themselves  of  such 
importance  as  to  make  the  total  losses  greater  than  in 
the  previous  example.  This,  however,  is  by  no  means 
necessarily  the  case  when  iron  is  introduced  in  the  mag- 


CAPACITY  55 

netic  circuit.  The  assumptions  made  for  the  purpose  of 
drawing  the  above  diagrams  are  arbitrary,  but  they  serve 
the  purpose  of  showing  in  what  manner  the  losses  in  the 
iron  core  appear  in  the  supply  circuit  and  affect  the 
magnitude  and  phase  displacement  of  the  total  current. 

25.  Capacity. — The  effects  of  electrostatic  capacity 
are  of  far  more  importance  in  alternating-current  than 
in  continuous-current  work,  and  we  shall  briefly  examine 
the  conditions  arising  out  of  the  introduction  of  a  con- 
denser in  a  circuit  carrying  an  alternating  current. 

The  chief  difference,  in  this  respect,  between  a  con- 
tinuous—or unidirectional — and  an  alternating  current 
is  that  the  former,  after  charging  the  condenser,  will  be 
absolutely  interrupted,  while  the  latter  will  continue  tcr 
flow  to  an  extent  depending  upon  the  capacity  of  the 
condenser,  the  pressure,  and  the  frequency  (not  to  mention 
the  wave  form,  which  must  also  be  taken  into  account). 

Any  arrangement  of  two  conductors  of  electricity 
separated  by  an  insulator  forms  a  condenser,  of  which 
the  capacity  will  be  large  or  small  depending  upon  the 
nearness,  or  otherwise,  of  the  conductors,  and  the  nature 
of  the  dielectric  between  them. 

The  effects  of  electrostatic  capacity  in  the  case  of 
conductors  supported  on  poles  at  a  reasonable  distance 
from  the  ground  are  very  small,  even  on  a  long  trans- 
mission line,  provided  the  frequency  is  low;  but  further 
reference  will  be  made  to  this  particular  case  when 
treating  of  the  transmission  of  power  by  polyphase 
currents.  The  capacity  of  an  underground  cable, 
especially  when  insulated  with  rubber,  is  a  much  more 
serious  matter.  The  capacity  per  mile  of  any  particular 
make  and  size  of  cable  will  always  be  furnished  by  the 
manufacturer,  and  the  total  capacity  of  any  feeder  or 


SELF-INDUCTION   AND   CAPACITY 


distributing  system  can,  therefore,  be  readily  calculated 
when  the  length  of  the  different  sizes  of  cable  is  known. 

26.  Current  Flow  through  a  Condenser. — If  two 

ballistic  galvanometers  are  connected  to  the  terminals  of 
a  condenser,  in  series  with  a  battery  or  other  source  of 
continuous  E.M.F.,  as  shown  in  Fig.  21,  it  will  be  found, 
on  depressing  the  key  K,  that  the  swing  of  both  galva- 
nometers is  such  as  to  indicate  that  the  same  quantity  of 
electricity  which  passed  into  one  set  of  plates  has  passed 
out  of  the  other  set  of  plates.  But,  nevertheless,  the 
condenser  is  now  charged,  and  if  the  key  Kx  be  depressed 


I  I  I 


K, 


FIG.  21. 

(after  releasing  the  key  K),  the  same  quantity  of  electricity 
will  flow  through  the  circuit  as  when  the  charging  key 
was  depressed,  only  the  flow  will  be  in  an  opposite 
direction,  as  indicated  by  the  swing  of  both  galvanometers 
being  approximately  the  same  as  before,  but  in  the 
reverse  direction.* 

*  A  charged  condenser  should  be  considered,  not  as  containing 
a  store  of  electricity,  but  a  store  of  electric  energy,  which  may  be 
used  for  doing  useful  work  by  joining  the  terminals  of  the  condenser 
through  an  electric  circuit.  In  this  respect  it  may  with  advantage 
be  compared  with  a  deflected  spring  which,  so  long  as  it  is  kept  in 
a  state  of  strain,  has  a  certain  capacity  for  doing  work. 


CURRENT  FLOW  THROUGH  A  CONDENSER   57 

Bearing  these  facts  in  mind,  it  will  not  be  difficult  to 
understand  that  an  alternating  current  will  have  the 
property  of  flowing  through  a  condenser,  and  the  following 
remarks  should  assist  the  reader  in  forming  a  mental 
picture  of  what  takes  place, 

In  Fig.  22,  A  is  an  alternator  and  C  a  condenser,  which, 
in  practice,  might  be  a  long  concentric  cable  with  the  two 
conductors  disconnected  and  well  insulated  at  the  distant 
end 

So  long  as  the  alternating  current  is  flowing  in  a 
positive  direction,  the  condenser  is  being  charged,  but  as 


FIG.  22. 

soon  as  ever  the  current  reverses,  the  condenser  begins 
to  discharge.  The  maximum  charge  of  the  condenser — 
which  corresponds  with  the  maximum  value  of  the 
condenser  E.M.F. — will,  therefore,  of  necessity  occur  at 
the  instant  when  the  current  is  changing  in  direction. 

In  Fig.  23,  the  curve  I  represents  the  current  flowing 
through  the  condenser,  and  the  dotted  curve  Q  represents 
the  charge  (in  coulombs),  which,  it  will  be  noticed,  has 
been  steadily  growing  from  its  negative  maximum  value 
to  its  positive  maximum  value  (d)  during  the  whole  time 


58  SELF-INDUCTION    AND   CAPACITY 

the  current  has  been  flowing  in  a  positive  direction.  This 
curve  Q  will  therefore  lag  behind  the  current  curve  by 
exactly  a  quarter  of  a  period. 

Next,  as  regards  the  condenser  E.M.F. ;  it  is  evident 
that  this  will  always  be  such  as  to  oppose  the  applied 


•^j.--'  Q 


FIG.  23. 

E.M.F.,  or,    n   other  words,    it   will   tend   to  expel  the 
charge. 

We  may,  therefore,  draw  the  curve  Ex  exactly  opposite 
to  Q,  the  length  of  the  ordinate  at  every  point  being 
equal  to 

quantity  (or  charge  of  electricity) 
capacity  of  condenser 


CURRENT  FLOW  THROUGH  A  CONDENSER   59 

If  we  neglect  the  resistance  of  generator  and  leads, 
and  also  certain  losses  which  occur  in  the  condenser 
itself,  the  curve  Ejin  Fig.  23  —  i.e.,  the  potential  difference 
at  condenser  terminals  —  will  be  exactly  equal  and 
opposite  to  the  alternator  E.M.F.  Thus,  the  condenser 
E.M.F.  is  exactly  a  quarter  of  a  period  in  advance  of  the 
current  ;  and  this  is  interesting  when  it  is  remembered 
that  the  E.M.F.  of  self-induction  always  lags  behind  the 
current  by  the  same  interval. 

This  suggests  the  possibility  of  counteracting  the 
effects  of  self-induction  by  the  introduction  of  capacity, 
or  vice  versa  ;  and,  indeed,  the  one  is  sometimes  used  in 
practice  for  the  purpose  of  partially  neutralising  the 
effect  of  the  other,  with  moderate  success  ;  variations  in 
the  frequency  or  any  slight  modification  in  the  wave  form 
of  the  applied  E.M.F.,  such  as  will  occur  as  the  load  on 
the  generator  is  varied,  will  considerably  interfere  with 
the  balancing  action  of  a  condenser  and  choking  coil. 

With  regard  to  the  magnitude  of  the  condenser  current; 
let  C  be  the  capacity  in  farads  of  a  condenser  with  an 
alternating  E.M.F.  of  maximum  value  Emax.  volts  between 
the  two  sets  of  plates.  The  total  charge  in  coulombs, 
at  the  end  of  one  half-period,  will  be  C  x  Emax.  But 
quantity  =  current  x  time  ;  hence  the  maximum  charge  in 
coulombs  is 


where  Im  is  the  mean  or  average  value  of  the  current, 
and  JL  is  the  time,  in  seconds,  during  which  the  current 

changes  from  maximum  to  zero  value.  Thus,  by  re- 
ferring to  Fig.  23,  it  will  be  seen  that,  while  the  current 
falls  from  its  maximum  at  a  to  zero  value  at  b,  the  charge 


60  SELF-INDUCTION   AND   CAPACITY 

has  grown  from  zero  value  to  its  maximum  value  at  d. 
We  can  therefore  write 


or 

Ira=4/CEmax. 

The  quantity  we  generally  require  to  know  being  the 
R.M.S.  value  of  the  current,  it  is  necessary  to  multiply 

the  above  quantity  by  the  form  factor,  or  ratio™tual  v*lue 

mean  value 

of  the  current.     If  the  applied  E.M.F.  is  sinusoidal,  the 
current  wave  will  also  be  a  sine  curve,  and  the  form 

factor  is  then 


T    _2  7T 

c~7? 

If  E  stands  for  the  virtual  or  R.M.S.  value  of  the 
E.M.F.  (obtained  by  dividing  the  maximum  value  by 
v^),  the  formula  becomes, 

IC=27T/C    E. 

The  capacity  is  usually  stated  in  microfarads,  and  if 
C  is  so  expressed,  the  right-hand  side  of  the  equation  must 
be  divided  by  1,000,000.* 

*  It  is  important  to  note  that  the  capacity  current  will  depend  con- 
siderably upon  the  wave  form  of  the  applied  potential  difference. 
Thus,  in  place  of  the  multiplier  2  TT  (or  6'283),  in  the  formula  for  a 
simple  harmonic  variation — or  sine  wave — we  should  have  to  substi- 
tute 4  /y/3  (or  6-928)  if  the  wave  were  of  a  triangular  shape,  with 
straight  sides,  and  this  corresponds  to  an  increase  of  10  per  cent, 
in  the  amount  of  the  capacity  current.  A  peaked  form  of  wave  is, 
therefore,  to  be  avoided  if  it  is  desirable  to  keep  down  the  capacity 


CURRENT  FLOW  THROUGH  A  CONDENSER   6l 

In  very  long  underground  feeders,  the  current  flowing 
into  the  cable  when  there  is  no  connection  at  the  distant 
end  is  sometimes  considerable,  especially  when  the  cables 
are  insulated  with  rubber,  and  the  frequency  and  pressure 
are  high.  This  is  the  capacity  current,  and  its  effect  in 
increasing  the  total  current  is  readily  ascertained  when  it 
is  remembered  that  this  component  of  the  total  current 
will  be  90  time-degrees  out  of  phase  with  the  useful 
current  delivered  at  the  distant  end  of  the  feeder,  if  we 
assume  the  load  to  be  non-inductive. 

Thus,  if 

I  =  the  ingoing  current ; 

I^the  condenser  current  (say  10  amperes); 

I2  =  the  outgoing  current  (say  60  amperes), 

we  may  write 

I  =  Vlj2  + 122  =  60^82  amperes, 

which  is  very  little  greater  than  the  outgoing  current; 
and  the  example  illustrates  the  fact  of  the  capacity 
current  becoming  of  less  and  less  importance  as  the  load 
on  the  feeder  increases. 


current  ;  but  since  the  wave  form  of  all  alternators  changes,  to  a 
certain  extent,  as  the  load  comes  on,  there  will,  in  any  case,  be  a 
variation  in  the  capacity  current  depending  upon  the  load,  and  this 
variation  may  even  amount  to  as  much  as  50  per  cent.  It  is,  how- 
ever, the  capacity  current  at  light  loads  which  is  of  the  greatest  im- 
portance in  practice,  because  at  times  of  heavy  load  its  effects  are 
not  felt  to  anything  like  the  same  extent.  Should  the  E.M.F.  wave 
have  several  peaks—  i.e.,  more  than  one  maximum  value  per  half 
wave — this  would  be  quite  the  worst  form  of  wave  as  regards  the 
capacity  current,  which  might  easily  be  doubled,  solely  on  account 
of  this  peculiarity  in  the  shape  of  the  wave. 


62 


SELF-INDUCTION    AND   CAPACITY 


Perhaps  one  of  the  most  extensive  systems  of  con- 
centric, rubber-insulated,  underground  cables  is  that  of 
the  Societe  Anonyme  Beige  Eclairage  Electrique  de 
Saint  Petersbourg.  There  are  about  240  kilometres  of 
cable  in  St.  Petersburg  forming  the  high-tension  network 
of  this  company's  mains.  The  charging  current,  for  the 
frequency  of  42^5  and  a  pressure  of  2,000  volts,  is  about 
47  amperes;  but  there  are  nearly  1,000  transformers  on 
the  circuits,  taking  a  total  magnetising  current  of  about 

n  Resistance  E2 


9 


8 


E, 


FIG.  24. 


250  amperes,  which  more  than  balances  the  abnormally 
large  capacity  current. 

27.  Condensive  Reactance.  —  Just  as  inductive 
reactance  can  be  expressed  in  ohms  (see  article  1 7,  p.  40), 
so  also  can  we  express  the  reactance  due  to  the  capacity 
of  a  circuit,  provided  always  that  the  connection  between 
frequency  and  reactance  is  not  lost  sight  of. 

Reactance  is  a  quantity  which,  when  multiplied  by  the 
current,  will  give  that  component  of  the  total  E.M.F.  in 
the  circuit  which  is  necessary  to  balance  the  counter 
E.M.F.  of  self-induction  or  of  capacity,  as  the  case  may 
be.  It  may  therefore  be  thought  of  as  the  counter 


CONDENSIVE   REACTANCE  63 

E.M.F.  in  the  circuit  when  unit  current  is  flowing  ;  and 
when  represented  by  vectors,  it  must  always  be  drawn 
at  right  angles  to  the  current  vector.  But  whereas  the 
vector  representing  the  balancing  component  of  the 
inductive  reactance  is  drawn  in  advance  of  the  current 
vector  (see  Figs.  17  and  18),  the  corresponding  vector 
representing  condensive  reactance  must  be  drawn  in  the 
direction  of  retardation  —  i.e.,  90  time-degrees  behind  the 
direction  of  the  current  vector. 

In  Fig.  24,  O  E2  is  the  resistance  of  the  circuit,  which 
may  also  be  considered  as  the  voltage  component  in 
phase  with  the  current,  divided  by  the  current. 

E2  E  is  the  condensive  reactance,  of  which  the  value  in 
ohms,  on  the  sine  wave  assumption,  is  — 


since  it  may  be  considered  as  the  voltage  component 
required  'to  balance  the  condenser  E.M.F.,  divided  by  the 
current. 

O  E  is  the  impedance,  of  which  the  value,  in  ohms,  is  — 


and  it  may  be  considered  as  the  total  E.M.F.  impressed 
on  the  circuit,  divided  by  the  current. 

The  reason  why  condensive  reactance  may  be  con- 
sidered as  the  exact  opposite  of  inductive  reactance  should 
now  be  clear  to  the  reader  ;  because  if  the  circuit  were 
supposed  to  have  inductance  as  well  as  capacity,  the 
reactance  due  to  the  choke  coil  effect  would  be  a  con- 
tinuation of  the  line  E  E2  of  Fig.  24,  but  it  would  be 
drawn  above  the  line  O  E2  ;  and  the  complete  formula  for 


64  SELF-INDUCTION   AND   CAPACITY 

the  impedance  of  a  circuit,  taking  into  account  resistance, 
self-induction,  and  capacity,  is — 

Z- I 


— where  w  =  2  TT  f,  on  the  sine  wave  assumption. 


CHAPTER  III 

POLYPHASE    CURRENTS — GENERAL    PRINCIPLES    AND 
SYNCHRONOUS    GENERATORS 

28.  WHEN  single-phase  alternating  currents  were  first 
used,  on  a  large  scale,  for  incandescent  electric  lighting, 
no  satisfactory  single-phase  motors  were  obtainable.  One 
of  the  difficulties  in  the  way  of  producing  commercial 
alternating-current  motors  was,  no  doubt,  the  high 
frequency  (about  100  in  this  country)  which  had  been 
adopted  for  the  lighting  circuits ;  but,  apart  from  this,  the 
problem  of  designing  single-phase  motors  capable  of 
starting  under  load'is  not  a  simple  one.  The  introduction 
of  polyphase  currents,  by  the  aid  of  which  a  rotating 
magnetic  field  could  be  obtained  from  stationary  windings, 
completely  solved  the  problem  of  the  alternating-current 
motor. 

The  idea  of  producing  a  rotating  magnetic  field  by 
means  of  polyphase  currents  may  be  said  to  date  as  far 
back  as  1879,  when  Walter  Baily  read  a  paper  before 
the  Physical  Society,  entitled  "  A  Mode  of  Producing 
Arago's  Rotation ";  but,  although  several  patents  were 
taken  out  during  the  ensuing  years,  nothing  serious  was 
done  in  this  connection  until  1891,  when  M.  von  Dolivo- 
Dobrowolsky  and  Mr.  C.  E.  L.  Brown  put  down  their 
complete  scheme  of  three-phase  power  transmission  from 

65  5 


66 


POLYPHASE   CURRENTS 


Lauffen  to  Frankfurt.     After  this,  the  use  of  polyphase 
motors  extended  rapidly. 

Definition. — A  polyphase  current  may  be  defined  as  a 
combination  of  two  or  more  alternating  currents,  all  of 
the  same  periodicity,  but  having  certain  phase  differences 
between  them. 

29.  Production  of  Rotary  Field. — In  Fig.  25  we 
have  a  representation  of  a  two-phase  current,  the  phase 


FIG.  25. 

angle  of  which  is  90  degrees.  It  will  be  seen  that  this 
diagram  merely  represents  two  simple  harmonic  alter- 
nating currents,  A  and  B,  of  the  same  frequency  and 
the  same  amplitude,  but  differing  in  phase  by  a  quarter 
of  a  period. 

Let  us  imagine  these  two  currents  to  be  flowing  re- 
spectively in  two  coils,  A  and  B,  set  at  right  angles  to 
each  other,  in  the  manner  shown  in  Fig.  26.  We  shall 


PRODUCTION   OF  ROTARY   FIELD 


67 


further  assume  that  when  the  current  A  is  flowing  in  a 
positive  direction — i.e.,  when  it  is  measured  above  the 
datum  line,  Fig.  25 — it  will  produce  a  magnetic  field  in 
the  upward  direction  through  the  coil  A  ;  and  when  the 


FIG.  26 

current  B  is  positive,  it  will  produce  a  field  from  left  to 
right  through  the  coil  B. 

In  order  to  see  clearly  how  the  resultant  field,  due  to 
the  currents  in  the  two  coils,  revolves  round  the  centre, 
O  (Fig.  26),  let  us  start  at  the  instant  O  (Fig.  25).  The 
resultant  field  at  this  instant  is  evidently  O  R,  which 


68  POLYPHASE  CURRENTS 

is  the  vector  representation  of  the  field  due  to  the 
maximum  negative  value  of  the  current  B — the  value 
of  A,  at  this  instant,  being  zero. 

After  the  lapse  of  one-eighth  of  a  period,  the  two 
currents  (on  the  assumption  of  a  sine  curve  wave  form) 
are  exactly  equal,  but  A  will  be  positive  and  B  negative : 
the  resultant  field  will  be  O  S.  After  another  eighth  of 
a  period,  the  resultant  field  will  be  that  due  to  the 
maximum  positive  value  of  A ;  it  would  be  represented 
by  a  vector  equal  in  length  to  O  R,  but  drawn  in  a 
vertical  direction  from  the  centre,  O,  upwards.  At  the 
instant  t  (corresponding  to  the  7/i6ths  of  a  complete 
period)  A  =  £  m  and  B  =  t  n,  both  being  positive.  The 
combination  of  these  two  component  fields  gives  us  O  T 
as  the  resultant.  It  is  not  necessary  to  carry  the  con- 
struction any  further,  for  it  is  evident  that  the  combined 
effect  of  the  two  currents  will  be  to  produce  a  magnetic 
field  of  constant  intensity,  equal  to  that  due  to  the 
maximum  value  of  either  current,  revolving  round  O  at 
the  uniform  rate  of  one  revolution  in  the  time  of  one 
complete  period. 

Although  we  have  assumed  a  phase  difference  of  a 
quarter  period  between  the  two  currents,  and  an  angle  of 
go  degrees  between  the  axes  of  the  two  coils,  it  should 
be  stated  that  a  rotating  field  can  be  obtained  by  means 
of  two  phase  currents,  whatever  may  be  the  phase 
difference  between  them. 

The  necessary  condition  for  the  production  of  a 
uniformly  rotating  field  of  constant  strength  by  the  com- 
bination of  two  alternating  magnetic  fields  is  simply  that 
the  angle  which  the  positive  directions  of  the  alternating 
fields  make  with  each  other  must  be  the  supplement  of  the 
phase  angle. 

Thus,   if   we    had    assumed    a    phase    difference    of 


PRODUCTION   OF    ROTARY    FIELD  69 

100  degrees  instead  of  90  degrees  between  the  two 
currents  shown  in  Fig.  25,  we  should  have  found  it 
necessary  to  draw  the  coils  A  and  B  in  Fig-  26  with 
an  angle  of  (180-100)  degrees,  or  80  degrees  between 
them  instead  of  at  right  angles  to  each  other. 

With  regard  to  the  intensity  of  the  resulting  rotating 
field,  on  the  assumption  of  the  sine  law  of  variation,  this 
will  be  equal  to  the  maximum  value  of  either  of  the  two 
(equal)  alternating  fields,  multiplied  by  the  sine  of  the  phase 
angle. 

Although  the  proof  of  the  above  statements  is  simple, 
we  shall  content  ourselves  with  an  illustration,  which,  at 
the  same  time,  will  explain  the  use  of  another  diagram, 
sometimes  preferable  to  the  curves  of  Fig.  25,  for  repre- 
senting alternating  quantities  which  follow  the  simple 
harmonic  law  of  variation. 

We  shall  assume  the  phase  difference  between  the  two 
currents  to  be  one-third  of  a  period,  instead  of  one-quarter 
of  a  period,  as  in  the  previous  example. 

Draw  O  A  and  O  B,  in  Fig.  27,  to  represent  the 
maximum  values  of  the  two  alternating  magnetic  fields. 
These  vectors  must  be  drawn  with  an  angle  of  120  degrees 
between  them  to  represent  the  phase  difference  of  one- 
third  of  a  period. 

With  O  as  a  centre,  describe  the  dotted  circle  of  radius 
equal  to  O  A  or  O  B,  and  on  each  end  of  the  vertical 
diameter  describe  a  smaller  circle,  of  radius  equal  to 
half  O  A  or  O  B,  in  the  manner  shown  in  the  figure. 

Now  imagine  the  two  vectors  to  revolve  in  a  counter- 
clockwise direction — as  indicated  by  the  arrow — at  a  uniform 
rate  of  one  revolution  in  the  time  of  one  complete  period. 
The  length  of  either  vector,  measured  from  the  centre,  O, 
to  the  point  where  it  is  cut  by  one  of  the  inner  circles 
will  then  be  a  measure  of  the  field  intensity  due  to  the 


POLYPHASE   CURRENTS 


current  in  phase  A,  or  in  phase  B,  as  the  case  may  be,  at 
any  particular  instant  throughout  the  complete  cycle. 
Measurements  made  in  the  upper  circle  would  denote, 
say,  a  positive  direction  of  the  magnetic  field,  while 


measurements  made  in  the  lower  circle  would  indicate 
a  magnetic  flux  in  the  opposite,  or  negative,  direction. 

The  proof  of  the  above  statement  is  very  simple.  In 
the  first  place,  the  angle  O  P  S  of  the  triangle  OPS, 
constructed  within  a  semicircle  of  which  O  S  is  the 
diameter,  is  always  a  right  angle;  and  since  the  angle 


PRODUCTION  OF   ROTARY   FIELD  71 

P  S  O  is  equal  to  A  O  N,  we  see  that  O  P  is  always 
equal  to  O  S  sin  0,  or  O  A  sin  6,  which  is  the  same 
thing.  Now,  the  reason  why  the  simple  harmonic  curves 
as  shown  in  Fig.  25  are  also  called  sine  curves,  is  simply 
because  the  ordinates  of  such  curves  are  directly  propor- 
tional to  the  sine  of  the  time-angle  ;  and  it,  therefore, 
follows  that  the  portion,  O  P,  of  any  revolving  vector 
such  as  O  A  or  O  B,  which  is  contained  within  either 
of  the  two  small  circles,  will  be  a  measure  of  the  instan 
taneous  value  of  an  alternating  quantity  which  follows 
the  simple  harmonic  law  of  variation. 

Let  us  now  see  how  the  combination  of  the  two  alter- 
nating fields,  A  and  B — when  B  lags  behind  A  by  one- 
third  of  a  period  (120  degrees) — will  produce  a  rotating 
field  when  combined  at  an  angle  of  (180—120)  or  60 
degrees. 

In  Fig.  28,  the  two  coils  are  shown  at  an  angle  of 
60  degrees  to  each  other ;  and,  since  the  field  due  to  each 
coil  will  be  at  right  angles  to  the  plane  of  the  coil,  the 
field  component  due  to  coil  A  will  aways  lie  on  the 
diameter  10 — 4,  while  the  field  component  due  to  B  will 
lie  on  12 — 6. 

At  the  particular  instant  corresponding  to  the  position 
of  the  vectors  shown  in  Fig.  27,  the  resultant  field  will 
be  O — i,  obtained  by  compounding  the  vectors  O  c  and 
O  d.  After  an  interval  of  a  twelfth  of  a  period  (30  de- 
grees), the  two  components  will  be  equal,  but  of  the 
same  sign  as  before — i.e.,  A  will  still  be  positive,  and 
B  negative  :  the  resultant  field  is  O — 2,  of  which  the 
components  are  O — 12  and  O — 4.  After  another  twelfth 
of  a  period,  the  vector  A  (Fig.  27)  will  have  reached  its 
maximum  positive  value,  while  B  will  have  fallen  to 
half  its  maximum  negative  value  :  the  resultant  field  will 
be  O — 3,  obtained  by  compounding  O  e  and  O  /.  And  so 


72  POLYPHASE   CURRENTS 

on  :  it  will  be  found  that  the  resultant  field  will  revolve 
round  O  at  a  uniform  rate  of  one  revolution  per  period, 
and  it  will  have  a  constant  value  represented  by  the 
radius  of  the  dotted  circle  in  Fig.  28.  This  radius  is 
evidently  equal  to  O  c  x  sine  60  degrees  ;  that  is  to  say, 


B 


FIG.  28. 


the  intensity  of  the  uniform  rotatory  field  will — for  this 
particular  phase  difference — be  equal  to  the  maximum 
value  of  any  one  of  the  component  fields  multiplied  by 
the  sine  of  60  degrees. 

This  confirms  the  general  statement  previously  made, 


THREE-PHASE   CURRENTS 


73 


to  the  effect  that  the  resulting  rotary  field  is  equal  to  the 
maximum  value  of  either  of  the  two  equal  alternating 
fields,  multiplied  by  the  sine  of  the  phase  angle.  It  is  true 
that  the  phase  angle,  in  this  example,  is  120  degrees,  and 
not  60  degrees,  but  the  sine  of  1 20  degrees  is  the  same  as 
sine  (1.80  —  120)  degrees  or  60  degrees. 

30.  Three-Phase    Currents.  — It    has   been   shown 
how  a  rotating  field  may  be  produced  by  means  of  two- 


FIG.  29. 

phase  currents,  and  the  reader  should,  therefore,  have  no 
difficulty  in  understanding  that  the  same  results  can  be 
obtained  by  using  any  number  of  currents,  with  suitable 
phase  differences  between  them,  and  a  corresponding 
number  of  field  coils  arranged  with  the  proper  angular 
spacing  between  them. 

Fig.  29  shows  the   three   currents   of  a  three-phase 
system  with  a  phase  angle  between  them  of  120  degrees, 


74  POLYPHASE   CURRENTS 

or  one-third  of  a  complete  period.  It  is  not  proposed  to 
draw  any  more  diagrams  on  the  lines  of  Figs.  27  and  28 ; 
but  it  will  be  understood  that  we  have  merely  to  imagine 
a  third  vector  drawn  in  Fig.  27  in  the  direction  O  C,  and 
a  third  coil  in  Fig.  28  across  the  diameter  n — 5,  when  it 
will  be  found  that  the  combination  of  the  three  alternating 
fields  will  produce  a  uniform  rotating  field,  exactly  as  in 
the  case  of  the  two-phase  system;  the  only  difference 
being  that  the  intensity  of  the  rotating  field  will  now  be 
equal  to  1-5  times  the  amplitude  of  any  one  of  the  three 
(equal)  alternating  fields. 

In  practice  the  rotating  field  is  not  necessarily  of  con- 
stant strength,  neither  does  it  always  revolve  at  a  uniform 
rate.  Thus,  if  the  currents  in  the  coils  do  not  follow  the 
sine  law,  the  resultant  field  will  vary  in  amount. 

Again,  if  the  induction  in  the  iron  forming  the  mag- 
netic circuit  be  carried  to  high  values,  the  permeability 
may  be  far  from  constant,  and,  even  with  a  current 
following  the  simple  harmonic  law,  the  magnetism  might 
vary  in  a  very  different  manner.  It  is,  therefore, 
advisable  to  work  with  comparatively  low  inductions,  and 
to  use  generators  giving,  as  nearly  as  possible,  sine  curve 
E.M.F.  waves  under  all  conditions  of  load. 

The  effect  of  inaccurate  spacing  of  the  coils,  or  of 
unequal  currents  in  the  different  phases,  will  also  be  to 
produce  distortion  of  the  resulting  field  ;  but  the  remedy 
is  obvious. 

3 1 .  Utilisation  of  Rotary  Field. — Consider  a  small 
continuous-current  dynamo,  of  which  the  field  magnets 
are  fully  excited,  but  with  the  brushes  lifted  off  the 
commutator. 

If  there  is  no  fault  in  the  armature  winding,  it  will  be 
possible — by  means  of  a  lever  or  pulley  fixed  to  the  shaft 
— to  revolve  the  armature  by  hand;  because,  although 


UTILISATION   OF    ROTARY   FIELD  75 

E.M.F.s  will  be  generated  in  the  windings,  these  balance 
each  other,  and  there  will  be  no  circulating  current. 

If,  on  the  other  hand,  one  or  several  of  the  armature 
coils  are  short-circuited,  the  effect  will  be  the  same  as  if 
a  strong  brake  had  been  applied,  and  it  will  not  be  possible 
to  revolve  the  armature  by  hand,  except  at  a  very  slow 
rate.  If  only  one  coil  be  short-circuited,  it  will  be  possible 
to  move  the  armature  freely  so  long  as  the  shorted  con- 
ductors lie  in  the  gap  between  the  poles ;  but  as  soon  as 
they  enter  the  fringe  of  magnetism  under  the  pole  tips, 
the  braking  effect  will  be  felt.  Again,  if  all  the  coils  are 
shorted — a  result  which  may  be  attained  by  winding  a 
few  turns  of  bare  copper  wire  round  the  commutator — 
there  will  be  no  positions  of  the  armature  between  which 
it  can  be  freely  moved ;  but  the  magnetic  braking  action 
will  be  felt  uniformly  throughout  the  complete  revolution 
of  the  armature. 

It  is  evident  that  these  effects  are  due  to  the  currents 
generated  in  the  closed  coils  of  the  armature,  for  these 
will  always  flow  in  such  a  direction  as  to  oppose  the 
motion  of  the  conductors  through  the  field.  The  torque 
which  has  to  be  exerted  to  make  the  armature  revolve 
will  be  proportional  to  the  product  of  field  strength  and 
armature  ampere  turns  under  the  pole-pieces,  and  the 
power  required  to  revolve  the  armature  will  be  expended 
in  the  form  of  I2  R  losses  in  the  armature  windings. 

Let  us  now  suppose  that  the  whole  framework  of  the 
dynamo,  including  the  system  of  field  magnets,  instead 
of  being  bolted  to  solid  foundations,  is  free  to  revolve  on 
the  armature  spindle.  We  shall  not,  with  this  arrange- 
ment, experience  the  same  difficulty  in  revolving  the 
armature,  because,  if  the  system  of  (excited)  field 
magnets  be  perfectly  balanced,  and  the  friction  of  the 
bearings  be  small,  the  revolving  armature  will  draw  the 


76  POLYPHASE   CURRENTS 

field  magnets  round  with  it;  and,  indeed,  if  frictional 
losses  could  be  entirely  neglected,  the  armature  and  field 
would  be  magnetically  locked  together,  while  the  whole 
system  could  be  revolved  in  space  without  requiring  any 
expenditure  of  power. 

If  there  is  appreciable  frictional  resistance  to  the 
motion  of  the  system  of  field  magnets,  the  latter  will 
revolve  at  a  slightly  slower  speed  than  the  armature  ; 
that  is  to  say,  there  will  be  a  certain  slip,  or  relative  speed 
between  the  two  parts,  which  will  automatically  adjust 
itself  until  the  currents  generated  in  the  armature  con- 
ductors are  sufficient  to  produce  the  necessary  torque. 

Although  we  have  briefly  considered  the  case  of  a 
revolving  armature  dragging  the  field  with  it,  the  ar- 
rangement can  be  reversed,  and  if  the  system  of  field 
magnets  be  revolved,  the  short-circuited  armature  will 
be  dragged  round  at  approximately  the  same  speed :  the 
E.M.F.s  generated  in  the  armature  windings,  for  a  con- 
stant strength  of  field,  will  depend  upon  the  rate  at  which 
the  magnetic  lines  are  cut  by  the  armature  conductors — i.e., 
upon  the  relative  speed  of  field  and  armature. 

In  a  polyphase  motor,  the  armature,  or  rotor,  may 
consist  of  three  or  more  windings  closed  upon  themselves, 
or — as  is  frequently  the  case  in  small  machines — it  may 
be  of  the  squirrel-cage  type,  in  which  heavy  conductors, 
threaded  through  the  iron  stampings  near  the  periphery, 
are  joined  together  at  each  end  by  substantial  metal  rings. 
In  both  cases  the  effect  is  the  same  :  the  revolving  field- 
produced  in  the  manner  already  described,  by  the  windings 
on  the  stator — generates  E.M.F.s  in  the  short-circuited 
rotor  windings,  and  the  resulting  current  produces  a 
torque  tending  to  drag  the  armature  round  against  the 
resisting  forces ;  these  resisting  forces  being  made  up  of 
bearing  friction,  windage,  hysteresis,  and  eddy-current 


UTILISATION    OF   ROTARY   FIELD  77 

losses  when  running  light,  with  the  addition  of  the  torque 
due  to  the  load  on  the  motor  when  the  latter  is  doing 
useful  work. 

Although  the  theory  of  the  polyphase  motor  will  be 
treated  more  fully  in  a  later  chapter,  it  is  important  that 
the  reader  should  have  a  clear  mental  picture  of  the 
revolving  field  dragging  the  short-circuited  armature 
with  it.  Imagine  the  resistance  of  the  closed  armature 
windings  to  be  nil  ;  then,  whatever  may  be  the  load  on 
the  motor,  its  speed  would  be  absolutely  constant  ;  the 
armature  would  be  magnetically  locked  with  the  revolving 
field,  because  the  slightest  "  slip,"  or  difference  in  speed 
between  field  and  armature,  would  induce  infinitely 
great  currents  in  the  windings  of  the  latter,  and  instantly 
pull  it  into  step. 

In  practice  the  slip  does  not  exceed  5  per  cent,  and  it 
will  reach  its  greatest  value  when  the  motor  is  doing  its 
maximum  load.  It  may  be  asked,  what  does  this  5  per 
cent,  difference  of  speed  represent  ?  The  answer  is  that 
it  represents  power  lost  in  heating  the  armature  con- 
ductors. Thus,  if  the  field  revolves  at  1,000  revolutions 
per  minute  and  the  armature  at  960  revolutions,  and  if 
the  torque  at  this  particular  speed  is  100  statical  foot- 
pounds, we  have  : 

Horse-power  imparted  to  armature 

IOO  X  27TX  I,OOO 


=  19-05 
Horse-power  developed  by  armature 

_  100x2^x960 

33»o°° 

=  18-3 


78  POLYPHASE   CURRENTS 

hence,  horse-power  wasted  in  PR  losses  in  armature 
conductors 

=  75 

=  560  watts. 

A  thorough  understanding  of  how  the  current  in  the 
primary  circuit  (stator  windings)  grows  in  response  to 
the  demand  for  additional  power  as  the  load  on  the  rotor 
increases  is  not  possible  without  a  knowledge  of  the  main 
principles  underlying  the  working  of  alternate-current 
transformers :  we  shall,  therefore,  leave  the  more  de- 
tailed consideration  of  the  induction  motor  for  the  present, 
and  briefly  deal  with  the  question  of  polyphase  generators. 

32.  Polyphase  Generators. — It  is  not  intended,  in 
these  pages,  to  give  formulae  or  data  which  would  be  of 
use  to  those  engaged  in  the  design  of  polyphase  generators ; 
but  an  attempt  will  be  made  to  give  a  clear  explanation 
of  the  elementary  principles  involved  in  the  generation  of 
two-  or  three-phase  currents. 

In  nearly  all  large  alternators  or  polyphase  generators, 
the  system  of  field  magnets  revolves,  while  the  armature 
is  stationary.  With  this  arrangement  only  two  collecting 
rings  are  required,  for  conveying  the  exciting  current  to 
the  field  coils ;  the  armature  current  being  delivered  from 
stationary  terminals.  The  insulation  of  these  terminals, 
together  with  that  of  the  armature  coils,  is  more  easily 
carried  out  than  if  they  formed  part  of  the  movable 
portion  of  the  machine. 

For  low  pressures,  such  as  200  volts,  especially  if  the 
output  is  small,  a  design  of  machine  with  revolving  arma- 
ture will  generally  be  found  to  be  cheap  and  efficient. 

Nearly  all  alternating-current  generators  are  of  the 
multipolar  type— that  is  to  say,  they  are  provided  with 
more  than  one  pair  of  poles.  A  few  machines,  when 


POLYPHASE   GENERATORS  79 

driven  at  high  speeds  by  steam-turbines,  may  have  only 
two  poles,  but  these  are  the  exception. 

If  p  is  the  number  of  poles,  and  R   is   the  speed  in 
revolutions  per  minute,  then 


Frequency  =  _- - 


x   — 

2 


and  in  the  case  of  large  machines,  running  at  compara- 
tively low  speeds,  a  fairly  large  number  of  poles  will  be 
found  necessary  to  give  the  desired  frequency. 

Whatever  may  be  the  type  of  machine,  or  number  of 
poles,  we  may  consider  the  armature  conductors  to  be  cut 
by  the  magnetic  lines  in  the  manner  indicated  in  Fig.  30. 
Here  we  have  a  diagrammatic  representation  of  single- 
phase,  two-phase,  and  three-phase  windings.  In  each 
case,  the  system  of  alternate  pole-pieces  is  supposed  to 
move  across  the  armature  conductors  in  the  direction 
indicated  by  the  arrow.  It  will  be  noted  that  the  con- 
ductors of  each  phase  are  shown  connected  up  to  form  a 
simple  wave  winding ;  but  this  is  only  done  to  simplify 
the  diagram,  and  it  will  be  readily  understood  that  each 
coil  may  contain  a  number  of  turns,  attention  being  paid 
to  the  manner  of  its  connection  to  the  succeeding  coil,  in 
order  that  the  E.M.F.s  generated  in  the  various  coils 
shall  not  oppose  each  other. 

The  upper  diagram  shows  a  single  winding,  in  which 
an  alternating  E.M.F.  will  be  generated.  In  the  middle 
diagram  there  are  two  distinct  windings,  A  and  B,  so 
arranged  relatively  to  each  other  and  the  pole-pieces  that 
the  complete  cycle  of  E.M.F.  variations  induced  in  A 
will  also  be  induced  in  B,  but  after  an  interval  of  time 
represented  by  a  quarter  of  a  period.  This  diagram 
shows  the  positions  of  the  poles  at  the  instant  when  the 
E.M.F.  in  A  is  at  its  maximum,  while  in  B  it  is  passing 


8o 


POLYPHASE  CURRENTS 


through  zero  value.  From  these  two  windings  we  can, 
therefore,  obtain  two-phase  currents  with  a  phase  differ- 
ence of  90  degrees. 


FIG.  30. 

In  the  bottom  diagram,  the  arrangement  of  three 
windings  is  shown,  from  which  three-phase  currents 
can  be  obtained,  having  a  phase  angle  of  120  degrees 


E.M.F.    INDUCED   IN   GENERATOR   WINDINGS      51 

or  one-third  of  a  period  between  them.  It  will  be 
noticed  that,  at  the  instant  indicated  by  the  relative 
positions  of  coils  and  poles,  the  E.M.F.  in  A  is  at  its 
maximum,  while  in  B  and  C  it  is  of  a  smaller  value,  and 
in  the  opposite  direction. 

33.  E.M.F.  induced  in  Generator  Windings.— 
With  the  assistance  of  the  top  diagram  of  Fig.  30,  it  will 
readily  be  seen  that  the  total  number  of  magnetic  lines 
cut  by  any  one  armature  conductor  per  revolution  will  be 
NX/,  where  N  stands  for  the  total  magnetic  flux  passing 
into  the  armature  from  any  one  pole,  and  p  is  the  number 
of  poles. 

If  Z  be  the  total  number  of  active  conductors,  counted 
round  the  circumference  of  the  armature,  and  R  stands 
for  the  speed  in  revolutions  per  minute,  the  mean  E.M.F. 
in  volts  generated  in  any  one  winding,  if  all  the  conductors 
are  connected  in  series,  will  be 

R  i 

Em=N/x  Z  x  j- 


100,000,000 

We  are  not  generally  concerned  with  the  mean  value 
of  an  alternating  quantity ;  but  it  is  not  always  possible 
to  estimate  very  accurately  the  shape  of  the  induced 
E.M.F.  wave,  which  will  determine  the  relation  of  the 
v/mean  square  value  of  this  E.M.F.  to  its  mean  value. 
This  will  depend  upon  the  disposition  of  the  windings, 
the  shape  and  spacing  of  the  poles,  etc.  In  order  to 
obtain  the  actual  volts  at  terminals  on  open  circuit,  we 
shall  have  to  include  a  multiplier,  k,  which  will  depend 
upon  the  form  of  the  wave.  This  multiplier,  which  is  the 
ratio  of  the  \/niean  square  value  to  the  mean  value  of  the 
induced  E.M.F.,  is  called  the  form  factor.  The  general 

*  See  article  15,  Chapter  II. 


82  POLYPHASE    CURRENTS 

expression  for  the  open-circuit  terminal  pressure  therefore 
becomes  ; 

- 


If  the  E.M.F.  wave  is  a  sine  curve,  k  -  i-il. 

If  the  wave  were  rectangular  in  shape  (which  would 
not  be  possible  in  practice),  the  multiplier  k  would  be 
unity. 

Other  calculated  values  of  k  are  : 

For  triangular  shape  ......     k=ri6. 

For  semi-circle  or  semi-ellipse     A=  ro^.. 

34.  Connections  of  Polyphase  Armature  Wind- 
ings. —  Consider  the  armature  winding  of  an  ordinary 
continuous-current  two-pole  dynamo.  If  we  imagine  the 
commutator  of  such  a  machine  to  be  entirely  removed, 
the  winding  —  whether  the  armature  be  of  the  drum  or 
ring  type  —  will  be  continuous,  and  closed  upon  itself.  If 
the  armature  be  revolved  between  the  poles  of  separately 
excited  field  magnets,  there  will  be  no  circulating  current 
in  the  windings,  because  the  magnetism  which  passes  out 
of  the  armature  core  induces  an  E.M.F.  in  the  conductors 
exactly  opposite  and  equal  in  amount  to  that  induced  by 
the  entering  magnetism. 

If  we  now  connect  to  a  couple  of  slip  rings  two  points 
of  the  winding  from  the  opposite  ends  of  a  diameter,  the 
machine  will  be  capable  of  delivering  an  alternating  cur- 
rent. If  we  provide  three  slip  rings,  and  connect  them 
respectively  to  three  points  on  the  armature  winding 
distant  from  each  other  by  120  degrees,  the  machine  will 
become  a  three-phase  generator. 

In  this  manner,  polyphase  currents  of  any  number  of 
phases  can  be  obtained,  and  if  the  windings,  polar  spaces, 
etc.,  are  symmetrical,  there  will  be  no  circulating  current. 


POLYPHASE   ARMATURE   WINDINGS  83 

This  method  of  connecting  up  the  various  armature 
coils  of  a  polyphase  generator  is  known  as  the  mesh  con- 
nection. In  the  case  of  three-phase  currents  it  is  also 
referred  to  as  the  delta  connection. 

The  diagram,  Fig  31,  shows  the  three  equidistant 
tappings  from  armature  winding  to  slip  rings  required  to 
produce  three  phase  currents.  It  is  evident  that  the 
potential  difference  between  any  two  of  the  three  rings 


FIG.  31. 

will  be  the  same,  since  each  section  of  the  winding  has 
the  same  number  of  turns,  and  occupies  the  same  space 
on  the  periphery  of  the  armature  core.  Moreover,  the 
variations  in  the  induced  E.M.F.  will  occur  successively 
in  the  three  sections  at  intervals  corresponding  to  one- 
third  of  a  complete  period. 

Incandescent  lamps  may  be  connected  across  one  or 
all  three  phases,  as  shown  in  Fig.  32.  This  lamp  load  is 
practically  non-inductive ;  and  we  may,  therefore,  consider 


84 


POLYPHASE   CURRENTS 


the  current  to  be  in  phase  with  the  potential  difference 
across  the  lamp  terminals. 

Let  the  vectors  O  I1}  O  I2,  and  O  I3,  in  Fig.  33,  repre- 
sent the  loads  on  the  three  sections  of  the  three-phase 
generator.  They  are  drawn  at  an  angle  of  120  degrees 
to  each  other,  because — being  in  phase  with  their  re- 
spective E.M.F.s — they  must  necessarily  differ  in  phase 
by  a  third  of  a  period. 

Referring  to  Fig.  32,  we  see  that  the  resultant  currents, 


FIG.  32. 

at  the  three  terminals  of  the  generator,  which  flow  into 
the  external  circuit  are : 


At  the  terminal  A 


T  T 

A         X 


It  is,  therefore,  an  easy  matter  to  draw  the  vectors 
O  A,  O  B,  and  O  C  in  Fig.  33  ;  these  represent,  by  their 
length  and  phase  relations,  the  currents  in  the  three 
conductors  leading  to  the  lamp  load. 

It  will  be  found  that  any  one  of  these  three  vectors 


POLYPHASE   ARMATURE   WINDINGS  85 

always  balances  the  other  two ;  that  is  to  say,  any  one 
vector,  such  as  B,  will  be  found  to  be  exactly  equal  but 
opposite  in  direction  to  the  resultant  of  A  and  C.  That 
this  is  necessarily  the  case  is  evident  when  it  is  realised 
that,  at  every  instant,  the  total  of  all  currents  leaving  the 


FIG.  33. 

armature  must  be  equal  to  the  total  of  all  currents 
returning  to  the  armature,  which  is  merely  another  way 
of  saying  that  the  sum  of  the  three  currents,  A,  B,  and  C , 
must  be  zero. 

If  the  load  is  balanced,  which  will  be  the  case  if  there 


86  POLYPHASE   CURRENTS 

is  an  equal  number  of  lamps  on  each  of  the  three  sections, 
or  if  the  load  consists,  not  of  lamps,  but  of  induction 
motors,  then  the  three  currents,  lv  I2,  and  I3,  will  be 
equal,  and  the  currents  leaving  the  terminals  will  also  be 
equal,  but  greater  than  the  armature  currents  :  it  can 
easily  be  shown  that,  for  this  condition  of  a  balanced 
load,  any  one  of  the  line  currents  is  equal  to  the  current 
in  any  one  of  the  armature  sections,  multiplied  by  2  cos 
30  degrees,  or  by  ^3.  Thus 

O  A  =  I    V 


Nature  of  Circulating  Current  in  Practice.  —  Although 
there  is,  theoretically,  no  circulating  current  in  the  mesh 
connected  armature  windings  of  a  polyphase  generator, 
the  conditions  of  practice  are  such  that  perfect  balance 
is  rarely,  if  ever,  attained,  and  currents  will,  therefore, 
circulate  in  the  windings  in  addition  to  those  delivered  to 
the  mains.  Such  currents  do  not  necessarily  involve  any 
appreciable  loss  of  efficiency,  but  they  must  evidently 
increase,  to  a  certain  extent,  the  PR  losses  in  the 
armature  conductors. 

The  circulating  current  with  the  field  magnets  fully 
excited,  but  with  open  external  circuit,  would  be  a  small 
percentage  of  the  normal  full  -load  current  in  a  well- 
designed  machine  ;  at  the  same  time,  if  there  is  want  of 
symmetry  in  the  arrangement  of  armature  coils,  or  in  the 
spacing  and  shapes  of  the  pole-pieces,  this  circulating 
current  might  amount  in  practice  to  as  much  as  25  per 
cent,  of  the  full-load  current.  A  little  consideration  will 
make  it  clear  that  the  function  of  this  internal  armature 
current  is  to  maintain  the  potential  differences  at  the 
three  terminals  such  that  their  sum  is,  at  every  instant, 


POLYPHASE   ARMATURE   WINDINGS  87 

equal  to  zero.  The  direction  of  this  current  at  every 
instant  will  be  such  as  to  react  upon  the  pole-pieces,  and 
so  produce  the  necessary  correction  in  the  distribution  of 
the  magnetic  lines  entering  the  armature.  The  frequency 
of  the  circulating  current  will  be  three  times  that  of  the 
line  current,  because,  while  a  current  of  the  fundamental 
frequency  cannot  circulate  in  a  delta-connected  winding, 
the  third  harmonic  and  its  multiples  can  circulate  freely. 
One  interesting  conclusion  to  be  drawn  from  these  con- 
siderations is  that  the  amount  of  the  circulating  current 
will  depend  upon  the  strength  of  the  field,  or  on  the 
magnetic  induction.  In  other  words,  if  an  ammeter  were 
inserted  in  series  with  the  closed  armature  windings,  it 
would  be  found  to  indicate  an  approximately  steady 
reading  notwithstanding  large  variations  in  the  speed  of 
the  machine,  provided  the  excitation  of  the  field  magnets 
remained  constant.  If  the  speed  were  kept  constant,  the 
amount  of  circulating  current  would  be  approximately 
proportional  to  the  volts  at  terminals. 

Star  Connection  of  Three-Phase  Armature  Windings. — In 
practice  this  method  sometimes  has  advantages  over  the 
mesh  or  delta  connection.  It  consists  merely  in  joining 
together  the  starting  ends  of  the  three  armature  windings 
in  one  common  junction,  or  neutral  point,  and  taking  the 
finishing  ends  to  the  three  terminals.  It  is  not  necessary 
to  run  a  common  return  connection  back  from  the  load 
to  this  neutral  point ;  each  arm  of  the  star  connection 
forms  the  return  path  for  the  currents  flowing  in  the 
other  two  arms. 

This  method  of  connecting  up  the  coils  will  be  again 
referred  to  when  dealing  with  the  question  of  transmis- 
sion of  power  by  three-phase  currents ;  but  if  we  refer 
back  to  Fig.  33  and  imagine  the  vectors  O  I15  O  I2,  and 
O  L  to  stand,  not  for  the  currents  in  a  mesh-connected 


88 


POLYPHASE   CURRENTS 


armature,  but  for  the  E.M.F.s  in  the  three  sections  of 
the  star  connection,  then  the  vectors  O  A,  OB,  and  O  C 
will  represent  the  potential  differences  between  the  three 
terminals  of  the  generator. 

A  simpler  construction  is  shown  in  Fig.  34.  Here 
o  ev  o  e2,  and  o  ey  represent  the  E.M.F.s  in  the  three 
sections  of  the  star-connected  armature,  and  the  sides  of 
the  triangle  obtained  by  joining  the  ends  of  these  three 
vectors  correctly  represent — both  by  their  lengths  and 


their^directions — thejamounts  and  phase  relations  of  the 
pressures  as  measured  between  terminals. 

If,  as  is  usually  the  case,  the  three  armature  sections 
have  the  same  number  of  turns,  the  triangle  of  vectors 
representing  the  terminal  pressures  will  be  equilateral, 
and — as  explained  in  connection  with  the  resultant  current 
of  a  W£sA-connected  armature — the  volts  measured  across 
the  terminals  of  a  stay-connected  machine  will  be  Vj  times 
greater  than  the  volts  measured  across  any  one  of  the 
three  (equal)  armature  sections. 


POLYPHASE  ARMATURE   WINDINGS  89 

Except  for  the  possible  advantage  of  being  able  to 
earth  the  neutral  point  of  the  star-connected  winding, 
it  is  of  little  consequence  to  the  user  whether  his  machines 
are  star  or  mesh  connected :  he  is  merely  concerned  with 
the  volts  at  terminals,  and  not  with  the  combination  of 
windings  which  produces  this  voltage. 

So  far  as  the  wave  shape  of  the  terminal  voltage  is 
concerned,  it  may  be  pointed  out  that  this  is  not  neces- 
sarily the  same  as  the  wave  shape  of  the  E.M.F.  developed 
in  the  armature  windings.  Thus,  what  is  known  as  the 
third  harmonic,  and  all  multiples  of  the  third  harmonic, 
are  absent  from  the  voltage  measured  across  the  terminals 
of  a  star-connected  three-phase  generator.  By  the  third 
harmonic  is  meant  a  sine  wave  of  three  times  the  perio- 
dicity of  the  fundamental  sine  wave,  which,  when  super- 
imposed on  this  fundamental  wave,  produces  distortion  of 
the  wave  shape. 

A  voltmeter  placed  across  the  terminals  of  a  star- 
connected  generator  measures  the  sum  of  two  vector 
quantities  with  a  phase  difference  of  60  degrees  (see 
Fig.  33,  p.  85).  Now,  a  phase  displacement  of  60  degrees 
of  the  fundamental  wave  is  equivalent  to  a  phase  dis- 
placement of  60  x  3  =  1 80  degrees  of  the  third  harmonic  ; 
that  is  to  say,  the  third  harmonics  cancel  out  so  far  as 
their  effect  on  the  terminal  voltage  is  concerned.  The 
general  rule  is  that  the  nth  harmonic  and  its  multiples 
cannot  appear  in  the  terminal  voltage  of  a  star-connected 
polyphase  generator  of  n  phases.  The  same  arguments 
apply  to  the  line  current  of  a  w^-connected  polyphase 
generator:  the  nth  harmonic  of  the  current  wave  can 
circulate  only  in  the  armature  windings ;  it  cannot  make 
its  appearance  in  the  current  leaving  the  terminals  of  the 
machine. 

Two-Phase  Armature  Connections. — It  is  generally  advis- 


90  POLYPHASE   CURRENTS 

able  to  keep  the  two  circuits  of  a  two-phase  supply 
entirely  separate,  in  which  case  the  generator  would  be 
provided  with  four  terminals  and  there  would  be  four 
conductors  connecting  the  machine  to  the  load ;  each  of 
the  two  circuits  can  be  treated  as  a  single-phase  alter- 
nating-current supply,  and  loaded  independently  of  the 
other ;  at  the  same  time,  if  the  two  circuits  are  brought 
to  an  induction  motor  with  stator  coils  correctly  spaced, 
a  revolving  field  will  be  produced. 


FIG.  35. 


Except  when  transmitting  the  current  to  long  distances 
(in  which  case  certain  complications  arise),  a  saving  of 
copper  may  be  effected  by  having  one  common  return 
conductor  which  will  carry  a  total  current  equal  to  the 
vector  sum  of  the  two  outgoing  currents. 

If  the  amounts  of  the  two  outgoing  currents  are 
respectively  A  and  B,  the  return  current  will  not  be 
A+B,  because  the  two  component  currents  are  not  in 
phase  ;  it  will  be  equal  to  J A2  -f  B2  for  a  phase  difference 
of  90  degrees,  as  will  be  evident  from  a  glance  at  Fig.  35, 


REGULATION  OF  SYNCHRONOUS  GENERATORS  91 

Thus,  if — as  would  be  the  case  on  a  load  consisting 
of  induction  motors — the  currents  are  equal  in  the  two 
phases,  the  return  current,  in  the  common  conductor, 
will  be  only  */2  times  as  great  as  the  current  in  any  one 
phase. 

The  equivalent  to  the  delta  or  mesh  connection  of  the 
three-phase  generator  would  be  obtained  by  taking  four 
tappings  off  the  closed  armature  winding,  instead  of  three 
as  in  Fig.  31.  A  generator  with  windings  connected  in 
this  manner  may,  therefore,  be  considered  indifferently 
as  a  two-phase  or  a  four-phase  machine. 

35.   Regulation  of  Synchronous  Generators. — 

The  polyphase  generators  we  have  been  considering 
belong  to  the  class  known  as  synchronous  machines,  to 
distinguish  them  from  the  asynchronous  generators  which 
are  practically  polyphase  induction  motors  reversed — i.e., 
of  which  the  rotor  is  mechanically  driven.  This  latter 
type  of  machine  will  be  briefly  referred  to  after  the 
induction  motor  has  been  more  fully  considered. 

Since  the  synchronous  polyphase  generator  is  very 
similar  to  its  prototype,  the  single-phase  alternator,  the 
scope  of  this  book  will  only  permit  of  the  following  points 
being  treated  somewhat  superficially.  At  the  same  time, 
the  manner  in  which  the  regulation  of  an  alternator 
depends  upon  the  nature  of  the  external  load  is  sufficiently 
important  to  claim  our  attention. 

In  a  continuous-current  dynamo,  the  drop  in  pressure, 
at  constant  speed,  between  no  load  and  full  load,  is  due 
not  only  to  the  resistance  of  the  armature  windings,  but 
also  to  the  armature  reaction — i.e.,  to  the  distorting  and 
demagnetising  effects  of  the  armature  current.  The 
same  thing  occurs  in  a  polyphase  generator,  and,  if  the 
load  is  non-inductive  — that  is  to  say,  if  the  current  is  in 
phase  with  the  potential  difference  at  generator  terminals 


92  POLYPHASE   CURRENTS 

—the  machine  will  behave  much  as  a  D.C.  dynamo  in 
respect  to  the  question  of  voltage  drop. 

In  Fig.  36,  the  alternate  poles  are  supposed  to  be 
travelling  at  the  back  of  the  armature  coils,  from  right 
to  left.  The  rectangles  A,  B,  and  C  represent  three 
different  instantaneous  positions  of  an  armature  coil 
relatively  to  the  poles.  In  position  A  the  induced 
E.M.F.  is  at  its  maximum,  and  tends  to  send  a  current 
in  the  direction  of  the  small  arrows.  In  position  B  the 
E.M.F.  is  zero  (since  there  is  no  change  in  the  number 
of  magnetic  lines  passing  through  the  coil),  while  in 


Direction  of  Travel  of  Poles 
FIG.  36. 


position  C  the  E.M.F.  is  again  at  its  maximum,  but  in 
the  opposite  direction  to  what  it  was  in  position  A. 

If  the  load  is  non-inductive,  the  current  will  be  in 
phase  with  the  E.M.F.,  and,  although  it  will  tend  to 
distort  the  magnetic  field,  it  will  have  no  direct  effect 
in  either  strengthening  or  weakening  the  inducing  field 
as  a  whole.  Now  imagine  the  load  to  be  inductive  ;  the 
current  will  lag  behind  the  E.M.F.,  and  will,  therefore, 
reach  its  maximum  value  when  the  pole-pieces  have 
travelled  beyond  the  position  indicated  at  A.  The  result 


COMPOUNDING   SYNCHRONOUS   GENERATORS      93 

will  be  that  the  current  in  the  armature  coils  will  have 
a  demagnetising  effect  on  the  field  magnets.  We  have 
only  to  imagine  the  worst  possible  condition,  of  the 
current  lagging  90  degrees,  to  see  that  the  poles  will 
occupy  the  position  B  relatively  to  the  armature  coil 
when  the  current  in  the  latter  is  at  its  maximum,  and  the 
effect  of  such  a  current  will  evidently  be  to  weaken  the 
pole  which  is  opposite  the  coil,  thus  diminishing  the  total 
flux  and  causing  a  drop  of  pressure  at  the  terminals. 

If  the  current  is  in  advance  of  the  E.M.F. — which  may 
occur  on  a  circuit  of  large  capacity — the  effects  would  be 
exactly  opposite  to  those  obtained  with  a  lagging  current : 
the  poles  would  be  strengthened,  and  this  strengthening 
effect  might  even  be  sufficient  to  counteract  the  weaken- 
ing due  to  distortion,*  and  the  drop  due  to  armature 
resistance  and  reactance ;  the  result  being  that  the 
peculiar  nature  of  the  load  might  alone  be  sufficient  to 
maintain  constant  volts  at  the  terminals  of  the  machine. 

36.   Compounding    Synchronous    Generators.— 

The  foregoing  remarks  will  make  it  clear  that,  in  order 
successfully  to  compound  an  alternating-current  generator, 
the  increased  field  excitation  must  not  merely  depend  upon 
the  amount  of  the  armature  ampere  turns,  but  also  upon 
the  nature  of  the  load — i.e.,  on  the  power  factor — and 
whether  the  armature  current  is  leading  or  lagging.  In 
practice  the  necessity  of  dealing  with  the  question  of 
heavy  leading  currents  will  not  arise,  although  such 
currents  will  frequently  occur,  at  light  loads,  on  a  circuit 
having  large  capacity,  such  as  would  be  obtained  with 
an  extensive  system  of  underground  cables.  The  require- 
ments of  a  perfect  alternator,  to  give  constant  volts  under 

*  Which,  by  crowding  the  magnetic  lines  to  one  side  of  the  pole- 
pieces,  increases  the  magnetic  reluctance,  and  diminishes  the  total 
flux  through  the  armature. 


94  POLYPHASE  CURRENTS 

all  conditions  of  load,  when  driven  at  constant  speed, 
may  be  summed  up  as  follows : 

(1)  The    terminal    pressure    should    remain    constant 
notwithstanding   alterations  in  amount  of   the  armature 
current  or  in  the  power  factor  (within  practical  limits). 

(2)  The  compensating  windings  or  devices  should  be 
such  as  to  be  practically  instantaneous  in  their  operation, 
so  that  the  generator  may  respond  promptly  to  the  altered 
conditions  of  load. 

(3)  The  compensating  or  compound  devices  must  not 
interfere  with  the  successful  running  of  the  machines  in 
parallel. 

This  last  condition  may  be  disposed  of  at  once  by 
stating  that,  in  the  case  of  compounded  alternators,  it 
may  be  just  as  necessary  to  provide  an  equalising  bar 
and  equalising  connections  between  the  machines  as  when 
running  direct -current  compound  -  wound  dynamos  in 
parallel. 

With  regard  to  the  condition  (2),  this  would  appear  to 
remove  from  the  field  of  useful  competition  all  automatic 
devices,  the  principle  of  which  depends  upon  the  varia- 
tion of  the  field  current  by  the  cutting  in  or  out  of 
resistances.  It  should  be  pointed  out,  in  this  connection, 
that,  even  if  the  current  round  the  field  magnets  were  to 
alter  instantly  with  the  change  of  load,  it  might  yet  be 
a  matter  of  several  seconds  before  the  terminal  pressure 
should  regain  its  correct  value  owing  to  the  large  in- 
ductance of  the  field  circuit.* 

The  nearest  approach  to  perfect  regulation  in  alter- 
nators by  varying  the  resistance  in  series  with  the  field 
coils  is  to  be  found  in  the  Tirrell  regulator,  which  is 
much  used  on  the  continent  of  America.  Its  success 
is  due  largely  to  care  in  design  and  attention  to  minor 
*  See  article  13,  Chapter  II.,  p.  28. 


COMPOUNDING   SYNCHRONOUS   GENERATORS      95 

details,  such  as  the  elimination  of  destructive  sparking  at 
the  relay  contacts.  The  main  principle  is  simple,  and 
well  worth  a  brief  description  even  in  these  pages,  from 
which  detailed  description  of  commercial  apparatus  and 
auxiliary  devices  are  of  necessity  excluded.  By  means 
of  certain  magnets  and  solenoids,  any  fall  of  exciter 
voltage,  or  terminal  voltage  of  the  generator,  automati- 
cally short-circuits  a  very  large  section  of  the  field 
rheostat.  The  greatly  increased  field  current  raises  the 
alternator  voltage,  with  the  result  that  the  short  circuit 
on  the  rheostat  is  again  automatically  removed.  The 
operation  is  repeated  at  a  rapid  rate,  and  the  results 
obtained  are  hardly  comparable  with  the  necessarily 
sluggish  regulation  obtained  by  cutting  in  or  out  seg- 
ments of  the  field  regulating  switches  in  the  ordinary 
way,  whether  by  hand  or  by  automatically  controlled 
power. 

Probably  the  most  successful  compound  generator  on 
the  market  is  the  self  -  exciting  synchronous  machine 
devised  by  Mr.  Alexander  Heyland,  whose  work  in  con- 
nection with  induction  motors  and  asynchronous  gener- 
ators is  well  known.  It  is  not  proposed  to  describe  his 
method  in  these  pages,  as  this  cannot  be  done  in  a  few 
words;  but  it  may  be  stated  that,  not  only  the  amount 
of  the  armature  current,  but  also  the  power  factor  is  taken 
into  account,  and  actual  tests  have  shown  that,  even  on 
zero  power  factor,  the  pressure  at  terminals  will  remain 
almost  constant  for  any  current  up  to  the  full-load  current 
taken  from  the  machine.* 

*  The  reader  who'cares  to  go  further  into  this  matter  is  referred 
to  Mr,  Eborall's  articles  in  the  Electrician  of  July  3  and  10,  1903,  and 
July  15,  1904  ;  also  to  the  excellent  paper  on  compensated  alternate- 
current  generators  read  by  Mr.  Miles  Walker  before  the  Manchester 
Section  of  the  Institution  of  Electrical  Engineers,  November  29, 
1904. 


96  POLYPHASE   CURRENTS 

There  would  appear  to  be  a  large  field  for  the  in- 
genuity of  inventors  in  this  matter  of  self-compounding 
alternators,  but  it  is  questionable  whether  the  demand 
for  such  machines  justifies  the  increase  of  cost  and 
possible  unreliability  due  to  any  departure  from  the  sim- 
plicity of  the  ordinary  designs.  An  automatic  regulator 
external  to  the  machine,  such  as  the  one  above  described, 
appears  to  fulfil  the  requirements  satisfactorily,  notwith- 
standing the  time  lag  that  must  necessarily  accompany 
any  change  of  current  in  the  field  windings. 

37.  Parallel  Running  of  Alternators.  —  An  at- 
tempt will  be  made  briefly  to  state  the  reasons  which 
account  for  the  successful  parallel  running  of  machines 
of  the  type  we  have  been  considering ;  but  it  is  not 
intended  to  investigate  in  detail  the  actions  that  take 
place  under  various  conditions  of  driving  or  of  load. 

In  the  first  place,  it  must  be  recognised  that  the  syn- 
chronous alternator  is  a  reversible  machine — that  is  to 
say,  it  will  run  as  a  motor  when  supplied  with  power  from 
an  alternating  source.  The  reasons  which  account  for 
the  continuous-current  dynamo  being  a  reversible  machine 
should  be  well  known  to  every  reader  of  these  pages,  and 
it  should,  therefore,  be  merely  necessary  to  point  out  that 
when  an  alternator  is  used  as  a  motor,  exactly  the  same 
forces  come  into  play  between  armature  conductors  and 
field,  provided  the  armature  is  supplied  with  current  at  the 
correct  frequency,  and  that  the  revolving  portion  of  the  machine 
has  been  run  up  to  the  correct  speed,  known  as  the  speed  of 
synchronism. 

Imagine  an  alternator  running  as  a  generator  on  load 
at  a  definite  speed,  or  frequency.  The  torque  exerted  by 
the  prime  mover  is  required — as  in  the  case  of  a  dynamo 
— to  force  the  conductors  carrying  current  across  the 


PARALLEL  kUNttiNG  oF  ALTERNATORS      9? 

magnetic  field.*  If,  now,  we  suppose  the  prime  mover 
to  be  disconnected,  and,  before  the  generator  has  had 
time  to  slow  down,  we  connect  the  armature  terminals 
to  an  alternating-current  supply,  the  machine  will  con- 
tinue to  run  as  a  motor,  provided  the  frequency  of  the 
supply  is  correct — i.e.,  such  as  to  correspond  with  the 
speed  and  the  number  of  poles. 

The  current  will  flow  in  the  armature  coils  in  the 
reverse  direction,  for  corresponding  positions  of  armature 
coils  and  poles,  the  result  being  a  number  of  impulses 
tending  to  keep  the  machine  in  step.  If  load  is  thrown 
on  the  machine,  there  will  be  a  momentary  retardation, 
causing  a  slight  displacement  of  phase  between  the  back 
E.M.F.,  due  to  the  conductors  cutting  the  magnetic 
flux,  and  the  applied  potential  difference:  these  forces 
will  not  be  exactly  opposed  to  each  other,  and  the 
result  will  be  an  increase  of  current  which  will  be  suf- 
ficient to  keep  the  machines  in  step.  If  the  machine 
breaks  out  of  step,  which  might  occur,  for  instance,  on 
a  sudden  and  considerable  overload,  then,  since  its  speed 
is  no  longer  that  of  synchronism,  the  regular  succession 
of  impulses  is  replaced  by  a  number  of  irregular  impulses 
in  both  directions,  and  the  machine  will  quickly  come 
to  rest. 

Although  the  synchronous  alternating- current  motor  is 
an  excellent  machine  for  certain  purposes,  it  has  the  dis- 
advantage of  not  being  self-starting,  f  When  running,  it 

*  It  is  true  that  the  forces  exerted  in  the  case  of  a  single-phase 
alternator  are  intermittent,  being  of  the  nature  of  a  series  of  im- 
pulses, and  not  continuous  as  in  a  dynamo  or  a  polyphase  machine  ; 
but  this  does  not  alter  the  argument. 

f  The  synchronous  polyphase  motor  will  sometimes  be  self- 
starting  on  light  loads  owing  to  the  rotating  magnetic  field  of  the 
stator  currents  setting  up  eddy  currents  through  closed  paths  in  the 
pole  faces  of  the  field  magnets,  even  if  no  copper  coils  are  specially 

7 


98  POLYPHASE   CURRENTS 

has  one  definite  speed,  depending  upon  the  frequency  of 
the  supply  circuit. 

With  regard  to  the  parallel  running  of  synchronous 
generators,  so  far  as  general  principles  are  concerned, 
there  is  little  to  add  to  the  foregoing  remarks.  If  two  or 
more  machines  are  electrically  coupled  to  common  'bus 
bars,  and  if  any  one  of  them— owing  to  a  momentary 
slowing  of  its  prime  mover,  or  from  any  other  cause — 
lags,  only  for  an  instant,  behind  the  others,  there 
will  be  an  immediate  rush  of  current  from  the  'bus 
bars,  which  will  tend  to  pull  it  into  step.  Success- 
ful parallel  running  depends  less  upon  the  design  of 
the  alternators  than  of  the  engines ;  an  even  turning 
moment  is  essential.  Large  and  small  units  can  be  run 
in  parallel  with  the  greatest  ease  ;  and,  although  it  is 
desirable  that  all  the  machines  should  give  approximately 
the  same  shape  of  E.M.F.  wave,  this  is  not  essential. 
The  portion  of  the  total  load  taken  up  by  each  of  several 
generators  in  parallel  will  depend  simply  upon  the 
amount  of  power  developed  by  the  respective  prime 
movers.  There  is  also  a  correct  value  of  the  exciting 
current  for  each  of  the  several  units  corresponding  to 
a  given  load ;  otherwise  the  necessary  magnetic  flux  will 
have  to  be  produced  by  internal  circulating  armature 
currents  provided  by  the  other  machines.  It  is  true  that, 
before  switching  an  additional  machine  on  to  the  'bus 
bars,  it  has  to  be  run  up  to  the  speed  of  synchronism  ; 
but,  once  switched  in,  it  is  magnetically  locked  with  the 


provided  in  the  pole  shoes  for  this  purpose.  If  this  method  is 
adopted,  the  volts  impressed  on  the  stator  windings  should  be 
reduced  below  normal,  and  connections  between  the  (unexcited) 
field  coils  should  be  broken  during  the  period  of  starting,  otherwise 
there  is  danger  of  the  total  E.M.F.  induced  in  the  field  circuit 
being  excessive. 


SYNCHRONISERS  99 

other  machines,  and  to  those  who  have  had  no  experience 
of  alternators  running  in  parallel,  it  is  surprising  to  note 
how  firmly  each  unit  appears  to  be  gripped  by  the  other 
machines  and  held  in  step. 

38.  Synchronisers. — When  a  synchronous  motor  is 
run  up  to  speed  by  means  of  an  auxiliary  induction  type 
motor,  or  when  an  extra  generator  is  about  to  be  thrown 
in  parallel  with  other  generators  feeding  common  'bus 
bars,  it  is  necessary  to  know  that  the  speed  of  syn- 
chronism has  been  reached  before  closing  the  main 
switches.  The  simplest  form  of  synchroniser  is  an 
incandescent  lamp  connected  across  the  switch  terminals. 
The  arrangement  for  three-phase  machines  is  shown  in 
Fig.  37.  As  the  motor  is  run  up  to  speed,  with  its  field 
excited,  the  lamps  will  flicker,  being  brightest  at  the 
instant  when  the  phases  are  so  related  that  the  maximum 
amount  of  current  will  tend  to  pass  between  the  machines. 
As  the  speed  of  the  motor  approaches  synchronism,  the 
"beats"  of  the  lamps  will  become  slower  and  slower, 
until  two  or  three  seconds  can  be  counted  between  the 
successive  periods  of  brightness  and  darkness.  If  the 
switches  are  then  closed  at  an  instant  when  the  lamps 
are  dark,  there  will  be  no  abnormal  rush  of  current 
between  the  machines,  because  they  will  be  in  syn- 
chronism. The  motor  must  be  provided  with  a  volt- 
meter so  that  the  field  current  can  be  adjusted  to  give 
approximately  the  same  voltage  as  the  generator. 

The  above  explains  the  principle  of  the  synchroniser. 
There  are  many  modified  and  convenient  devices  on  the 
market  which  enable  the  operator  to  close  the  switches 
at  the  right  instant.  It  is  obvious  that  a  voltmeter  or 
its  equivalent  can  be  used  in  place  of  the  lamp;  and  in 
any  case,  for  the  higher  voltages,  the  synchronising 
device,  of  whatever  type,  would  ordinarily  be  connected 


IOO 


POLYPHASE   CURRENTS 


in  the  secondary  circuit  of  a  step-down  transformer.  In 
the  arrangement  referred  to— of  three  lamps  connected 
directly  between  terminals— it  is  not  necessary  to  join 
these  up  in  the  manner  indicated  in  Fig.  37.  They  can 
be  so  connected  that  the  correct  instant  for  closing  the 
switches  is  when  all  three  lamps  are  bright ;  or,  if  desired, 
an  unsymmetrical  arrangement  of  connections  can  be 
adopted  by  which  the  lamps  will  glow  in  succession. 
This  has  the  advantage  of  indicating  by  the  order  in  which 


B 


Motor 


FIG.  37. 

the  lamps  light  up  whether  the  incoming  machine  is  run- 
ning too  fast  or  too  slow.  If  the  reader  is  interested  in 
such  details,  he  should  be  able  to  follow  out  the  necessary 
changes  in  the  lamp  connections  himself. 

Although  the  incoming  machine  has  been  referred  to 
as  a  motor,  the  procedure  is  obviously  the  same  when 
paralleling  generators  to  meet  the  demand  for  additional 
machines  as  the  load  increases. 

39.  Output  of  Polyphase  Generators. — A  generator 
wound  to. give  two  or  more  currents  differing  in  phase  will 
have. a  larger  output  than  if  it  is  wound  to  give  only 


OUTPUT   OF    POLYPHASE   GENERATORS          IOI 

single-phase  currents.  It  is  not  proposed  to  study  and 
compare  the  various  methods  of  winding  armatures,  in 
order  to  show  that  the  polyphase  generator  has  a  greater 
output  than  the  single-phase  machine ;  but,  owing  mainly 
to  the  fact  that  the  whole  of  the  armature  surface  can  be 
covered  by  the  windings,  it  is  found  in  practice  that,  for 
the  same  type  and  dimensions,  a  machine  provided  with 


FIG.  38. 


windings  for  two-phase  currents  will  have  an  output 
about  30  per  cent,  greater  than  if  wound  for  single  -phase 
currents. 

If  single-phase  currents  are  required,  they  can  always 
be  obtained  from  polyphase  machines.  Thus,  a  two- 
phase  generator  would  work  as  a  single-phase  machine 
with  its  two  windings  coupled  in  series,  the  terminal 


102  POLYPHASE   CURRENTS 

pressure  being  >/2  times  the  pressure  per  phase  for  the 
same  speed  and  excitation.  The  output,  however,  would 
not  be  so  great,  because,  although  the  current,  for  the 
same  heating  effects,  would  be  the  same  as  before,  the 
terminal  pressure  is  not  doubled  by  connecting  in  series 
the  two  (out  of  phase)  windings. 

The  power  of  the  two-phase  machine  =  2  (E  x  I),  where 
E  and  I  stand  respectively  for  the  pressure  and  current 
per  phase,  the  load  being  assumed  non-inductive.  The 
power  of  the  same  machine  used  as  a  single-phase  alter- 
nator will  be  only  >/2  E  x  I  for  the  same  losses  in  the 
armature  coils,  or  about  30  per  cent.  less. 

In  the  case  of  a  three-phase  generator,  we  may  connect 
all  three  windings  in  series  in  the  manner  indicated  by 
the  vector  diagram  Fig.  38;  and,  if  the  E.M.F.  in  each 
section  is  e,qual  to  E  volts,  the  resultant  E.M.F.  will  be 
2  E.  The  output,  for  a  current  I  in  the  armature  con- 
ductors, will  be  2  E  x  I  instead  of  3  (E  x  I),  which  would 
be  the  output  for  the  same  heat  dissipation  in  armature 
coils  if  the  machine  were  connected  up  for  three-phase 
working. 

A  slightly  greater  output  as  a  single-phase  machine 
will  be  obtained  by  utilising  only  two  of  the  three  sections. 
The  resultant  E.M.F.  will  be  only  \/3  E,  as  shown  by  the 
dotted  vector  in  Fig.  38  ;  but  for  equal  I2  R  losses  in 
armature  windings  the  current  may  now  be 


\/3  E  x  \/3  I 
making  output          =  -       —  .... 

V2 

-     3    E  I 

\/2 
=    2'12  El. 


CHAPTER  IV 

MEASUREMENT    AND    CALCULATION    OF    POWER    ON 
POLYPHASE    CIRCUITS 

40.  IT  is  not  proposed  to  describe  in  detail  the  principles 
of  construction  of  the  various  types  of  wattmeters  used 
for  measuring  power  on  alternating-current  circuits.  The 
single-phase  wattmeter  may  be  considered  as  a  modified 
form  of  Siemens  dynamometer,  with  fixed  thick-wire  coil 
and  movable  fine-wire  coil.  The  fixed  coil  carries  the 
main  current,  while  the  movable  coil  is  connected  in 
series  with  a  large  non  -  inductive  resistance,  and  takes 
a  current  proportional  to  the  potential  difference.  The 
object  of  the  non-inductive  resistance  is  to  reduce  the 
"  time  constant "  *  of  the  shunt  circuit  to  a  negligible 

*  It  was  explained  in  article  13  how  the  time  required  for  the 
growth  or  decrease  of  the  current  depends  upon  the  self-induction 
of  the  circuit  ;  but  the  effects  of  self-induction  will  depend  very 
much  on  the  resistance  of  the  circuit.  If  the  resistance  is  only  high 
enough,  the  counter  E.M.F.  of  self-induction  will  become  a 
negligible  quantity  in  comparison  with  the  E.M.F.  required  to 
overcome  the  resistance,  and  the  time  constant  of  a  circuit  may  be 
denned  as  the  ratio  of  its  coefficient  of  self-induction  to  its  resist- 
ance, or 

time  constant  =  ^. 

It  follows  that,  in  a  circuit  of  which  the  time  constant  is  small,  the 
effects  of  self-induction  may  be  negligible  whatever  may  be  the 
actual  value  of  the  E.M.F.  induced  by  the  passage  of  unit  current- 

103 


104      MEASUREMENT  AND  CALCULATION  OF  POWER 

amount,  in  order  that  the  current  in  this  circuit  may 
always  be  in  phase  with  the  potential  difference  across 
the  terminals. 

If  I  is  the  current  in  the  fixed  coil,  and  IE  the  current 
in  the  movable  coil,  the  reading  of  the  wattmeter  will  be 
proportional  to  I  I,,  cos  0,  where  0  is  the  phase  angle 
between  the  two  currents  ;  and,  since  IE  is  strictly  pro- 
portional to  the  terminal  potential  difference,  the  instru- 


ment  will   indicate   the   true   power   of  a    single-phase 
alternating-current  circuit. 

It  should  be  mentioned  that,  even  when  the  wave 
shapes  are  irregular  and  dissimilar — in  which  case  the 
phase  angle  6  must  be  considered  as  the  displacement 
between  equivalent  sine  waves — the  wattmeter  will  still 
correctly  indicate  the  true  power  in  the  circuit.  The 
force  tending  to  produce  deflection  of  the  pointer  is  at 
every  instant  proportional  to  the  product  of  the  currents 


POWER   OF   TWO-PHASE   CIRCUIT 


105 


in  the  two  coils ;  and  since  the  inertia  of  the  moving  parts 
does  not  permit  of  the  pointer  following  the  changes  of 
power  which  occur  throughout  a  complete  cycle,  the 
steady  deflection  obtained  is  actually  an  indication  of  the 
average  value  of  the  power  in  the  circuit.  Thus,  even  in 
the  case  of  irregular-shaped  waves  as  in  Fig.  10,  p.  22, 


FIG.  40. 

the  wattmeter  measures  the  average  ordinate  of  the 
shaded  power  curve,  and  automatically  subtracts  the 
negative  portions  of  the  power  from  the  positive  portions. 

41.  Power    of  Two -Phase   Circuit. — If   the  two 

circuits  are  separate — i.e.,  if  the  distribution  is  by  means 
pf  four  wires — two  wattmeters  would  be  required — one  in 


1O6      MEASUREMENT  AND  CALCULATION  OF  POWER 

each  circuit — and  the  sum  of  their  readings  would  give 
the  total  power  of  the  two  phases. 

If  the  load  is  balanced,  as  will  be  the  case  if  it  consists 
of  induction  motors  only,  the  pressure,  current,  and  angle 
of  lag  in  the  two  phases  will  be  the  same,  and  one  watt- 
meter will  suffice  ;  it  is  merely  necessary  to  double  its 
reading  to  obtain  the  total  power  of  the  two-phase  circuit. 

If  there  is  a  common  return  conductor  for  the  two 
phases,  one  wattmeter  will  also  suffice,  provided  the  load 
is  balanced.  If  the  distribution  of  load  is  unequal,  two 
wattmeters  are  required,  as  shown  in  Fig.  39. 

The  vector  diagram,  corresponding  to  Fig.  39,  for  a 
partly  inductive  unbalanced  load  has  been  drawn  in 
Fig.  40. 

Here  the  vectors  O  Ex  and  O  E2,  drawn  at  right 
angles  to  each  other,  and  equal  in  length,  represent  the 
pressures  Ex  and  E2,  with  a  phase  difference  of  90  degrees. 

The  vector  O  ll  shows  the  current  Ij  lagging  behind 
its  E.M.F.  by  an  angle  6,  while  O  I2  represents  the 
current  in  the  other  phase  lagging  behind  its  E.M.F.,  E2, 
by  an  angle  <£.  The  total  power  is  evidently  equal  to  the 
sum  of  O  Ex  x  O  m,  or  E±  Ix  cos  6  (measured  by  watt- 
meter Wj),  and  O  E2  x  O  n,  or  E2  I2  cos  <$>  (measured 
by  wattmeter  W2). 

42.  Power  in  Three-Phase  Circuit.— In  a  three- 
phase  system,  transmitting  power  by  means  of  three 
wires  only,  any  one  wire  may  be  considered  as  the  return 
conductor  for  the  other  two,  the  current  in  one  wire  being 
always  equal  to  the  sum  of  the  currents  in  the  other  two 
wires,  these  being,  of  course,  combined  in  the  usual  way, 
paying  due  attention  to  their  phase  relation. 

The  total  power  may,  therefore,  be  measured  by  means 
of  two  wattmeters  only,  exactly  as  in  the  case  of  a  two- 
phase  unequally  loaded  system.  The  wattmeters  are  in- 


POWER   IN   THREE-PHASE   CIRCUIT  IO? 

serted  in  any  two  of  the  three  conductors,  while  the  free 
ends  of  the  pressure  coils  are  connected — through  suitable 
non-inductive  resistances — to  the  third  conductor,  as 
shown  in  Fig.  41.  The  sum  of  the  two  wattmeter  read- 
ings will  correctly  indicate  the  total  three-phase  power, 
whether  the  load  be  balanced  or  not.  It  is  evidently  of 
no  consequence  whether  the  generator  windings,  or  the 
load,  be  star  or  mesh  connected,  provided  the  distribution  is 
by  three  wires  only  ;  but  if  the  generator  windings  are  star- 
connected,  and  a  fourth  conductor  is  run  between  the 
neutral  point  and  the  load,  for  the  purpose  of  connecting 


lighting  circuits  between  phases  and  neutral  conductor, 
then  three  wattmeters  will  be  necessary,  as  shown  in 
Fig.  42.  The  free  ends  of  the  three  shunt  windings  are 
connected  to  the  neutral  point,  or  return  conductor,  D, 
and  the  sum  of  the  readings  on  the  three  instruments  will 
evidently  be  the  total  power  on  balanced  or  unbalanced 
load — i.e.,  even  if  there  is  a  current  flowing  in  the  common 
conductor,  D. 

Balanced  Load. — When  the  load  consists  of  induction 
motors  only,  the  currents  in  the  three  conductors  will  be 
equal,  and  they  will  also  be  equally  displaced  in  phase 
relatively  to  the  corresponding  E.M.F.s. 


108      MEASUREMENT  AND  CALCULATION  OF  POWER 

There  are  two  cases  to  be  considered :  (i)  the  case  of 
star-connected  generators  with  neutral  point  available, 
and  (2)  the  case  of  mesh-connected  machines  in  which 
there  is  no  neutral  point  available.  The  arrangement 
shown  in  Fig.  42  is  applicable  to  case  (i),  if  we  imagine 
the  lamp  load  between  the  various  phases  and  the  common 
conductor,  D,  to  be  done  away  with.  The  load  being 
balanced,  the  currents  in  A,  B,  and  C  will  be  equal,  and 
equally  displaced  in  phase  relatively  to  the  respective 
E.M.F.s  measured  between  each  of  the  three  terminals 
and  the  neutral  point :  the  readings  of  the  three  watt- 


w, 


FIG.  42. 

meters,  W1?  W2,  and  W3,  will  be  identical,  and  any  one 
of  these  meters  will,  therefore,  suffice  to  measure  the 
load  on  a  balanced  three-phase  circuit ;  it  will  be  merely 
necessary  to  multiply  the  reading  by  three. 

Consider,  now,  case  (2),  in  which  the  neutral  point  is 
not  available.  The  arrangement  is  as  shown  in  Fig.  41, 
with  the  exception  that  one  of  the  wattmeters — say  W2 — 
is  supposed  to  be  entirely  dispensed  with.  The  remaining 
wattmeter,  Wr  with  the  free  end  of  the  fine-wire  coil 
connected  to  B  (as  shown  in  the  figure),  will  indicate  the 
quantity  E  I  cos  (30  degrees-  0),  where  0  is  the  angle  of 


POWER   IN    THREE-PHASE   CIRCUIT 


109 


lag  (cos  6  being  the  power  factor).  If,  now,  we  remove 
the  free  end  of  the  fine-wire  coil  from  B  to  C — leaving 
the  series  coil  in  circuit  with  the  conductor  A — the  instru- 
ment will  indicate  the  quantity  E  I  cos  (30  degrees  +  0). 
The  sum  of  these  two  expressions — after  the  necessary 
simplifications  have  been  effected — amounts  to  \/3  E  I 
cos  0,  which  is  the  expression  for  the  total  power,  where 
I  and  E  stand  respectively  for  the  current  in  any  one 
conductor  and  the  pressure  between  phases  of  a  balanced 
three-phase  system.  Thus,  even  when  the  neutral  point 
is  not  available,  the  sum  of  two  readings  taken  on  a 
single  wattmeter  will  give  us  the  true  power  of  the 
balanced  three-phase  circuit. 


FIG.  43. 

The  method  of  taking  two  readings  on  a  single  watt- 
meter would  not  be  convenient  in  practice,  neither  is  it 
necessary.  It  is  possible  to  combine  the  shunt  currents 
due  to  the  E.M.F.s  between  A  and  B,  and  A  and  C, 
respectively  (Fig.  41),  so  as  to  obtain  a  total  current  in 
the  fine-wire  coil  which  shall  give  the  required  reading 
on  the  one  wattmeter. 

Instead  of  changing  over  the  shunt  connection  of  the 
single  wattmeter  from  B  to  C  (Fig.  41),  it  is  mere!) 
necessary  to  provide  two  non-inductive  resistances  each 
of  v  ohms  (as  before),  and  connect  them  respectively  to 
the  two  conductors  in  which  there  is  no  wattmeter,  as 


1 10      MEASUREMENT  AND  CALCULATION  OF  POWER 

shown  in   Fig.   43.      The  wattmeter  will    then    directly 
indicate  the  total  power  of  the  balanced  circuit. 

This  arrangement  is  very  similar  to  an  alternative 
method  which  consists  in  producing  an  artificial  neutral 
point  by  means  of  a  star  resistance,  or  equivalent  arrange- 
ment in  which  a  choking  coil  is  used  :  it  is,  indeed,  only 
necessary  to  imagine  a  third  resistance,  of  r  ohms, 
inserted  between  the  fine-wire  coil  of  the  wattmeter  and 
the  point  O.  This — on  the  assumption  that  the  shunt- 
coil  resistance  is  negligible — would  have  the  effect  of 
reducing  the  current  in  the  shunt  coil  to  one-third  of  its 
previous  value,  and  the  wattmeter  would,  therefore, 
indicate  one-third  of  the  total  power,  exactly  as  if  it 
were  connected  to  the  neutral  point  of  a  star-connected 
generator  supplying  a  balanced  load. 

43.  Vector  Diagram  for  Calculating  Star  Resist- 
ances. —  The  diagram  Fig.  44  will  serve  to  indicate 
how  the  required  non-inductive  resistance  may  be  calcu- 
lated to  give  any  desired  shunt  current  through  the  watt- 
meter or  other  instrument  connected  in  one  phase  only. 

Let  a  b  c  be  the  E.M.F.  triangle — the  sides  of  which 
are  equal,  and  indicate  the  pressures  between  phases — 
while  c  C  represents  the  required  current  in  shunt  coil. 

The  first  and  all-important  point  to  bear  in  mind  in 
the  construction  of  such  diagrams  is  that  the  current,  C, 
leaving  any  one  terminal  must  exactly  equal  the  resultant 
of  the  other  two  currents,  A  and  B ;  in  other  words,  the 
sum  of  the  currents  at  the  three  terminals  is  always  equal 
to  zero.*  The  first  condition  to  be  fulfilled  in  Fig.  44  is, 
therefore,  that  the  three  current  vectors  a  A,  b  B,  and  c  C 
shall  form  a  closed  triangle. 

*  The  only  exception  being  the  case  of  a  star-connected  generator 
with  a  fourth  conductor  returning  to  the  neutral  point. 


DIAGRAM  FOR  CALCULATING  STAR  RESISTANCES     1 1 1 

Draw  the  dotted  line  B  A  parallel  to  the  side  b  a  of  the 
pressure  triangle,  and  at  a  distance  below  it  equal  to  half 
the  length  of  the  (known)  current  vector  c  C.  If,  now, 
we  join  any  point  such  as  Ol — lying  on  the  prolongation 
of  the  vector  c  C,  within  the  triangle  a  b  c — to  the  vertices 
a  and  b  of  the  pressure  triangle,  it  will  be  seen  that  the 
prolongation  of  the  lines  until  they  cross  the  dotted  line 


FIG.  44. 

B  A  will  give  us  the  current  vectors  a  i  and  b  i,  which, 
when  combined  together,  will  exactly  balance  c  C. 

It  will  be  seen  that  there  are  many  solutions  of  the 
problem,  but  the  required  ohms  in  the  three  arms  forming 
the  star  resistance  are  readily  calculated  for  any  position 
of  the  point  O. 

The  drop  of  pressure  in  each  arm  of  the  non-inductive 
resistance  is  represented  by  the  lengths  of  the  vectors 
O  a,  O  b,  and  O  c  (in  phase  with  the  corresponding 


112      MEASUREMENT  AND  CALCULATION  OF  POWER 

currents),  and  the  necessary  ohms  in  each  arm  can  readily 
be  calculated  by  dividing  the  lengths  of  these  pressure 
vectors  by  the  lengths  of  the  corresponding  current 
vectors — the  scale  to  be  used  for  the  pressure  vectors 
being,  of  course,  the  same  as  used  for  drawing  the 
triangle  a  b  c.* 

The  total  watts  lost  in  the  star  resistance  will  be  squal 
to  the  sum  of  the  I2  v  losses  in  the  various  arms  of  the 
star.  It  will  be  found  that,  if  c  C  is  kept  constant,  these 
losses  increase  as  the  point  O  is  taken  lower  down. 
They  will  be  greater  for  the  system  of  vectors  drawn 
from  the  point  O3  than  for  that  which  has  for  the  common 
point  O2  (the  centre  of  the  triangle);  but  the  most 
economical  arrangement  consists  in  doing  away  with  the 
resistance  in  series  with  the  current  C.  This  brings  the 
point  O  to  coincide  with  c,  and  makes  the  total  power 
absorbed  by  the  resistances  only  two-thirds  of  what  it 
would  be  if  a  star  resistance  with  three  equal  arms  were 
used.  It  will  be  observed  that  the  connections  in  this 
case  would  be  exactly  as  shown  in  Fig.  43. 

44.  Power  Factor  of  Three-Phase  Circuit. — The 

power  factor  of  any  circuit  carrying  an  alternating  current 

may  be  denned  as  the  ratio      true  ^owev    .     In  the  case 

apparent  power 

*  It  should  be  clearly  understood  that  the  sides  of  the  triangle  a  b  c 
represent,  by  their  length  and  direction,  the  magnitude  and  phase 
relations  of  the  pressures  measured  between  conductors,  while  the  star 
O  a,  Ob,  O  c,  represents  the  equivalent  system  of  vectors  all 
radiating  from  a  common  point  O.  That  such  a  system  of  star 
vector's  is  equivalent  to  the  triangle  of  vectors  is  proved  by  the 
fact  that  a  b  is  (vectorially)  equal  to  O  b  -  O  a.  Similarly  of  the 
other  vectors : 

be  =  O  c  -  O  b 
and' 

<?  a  =  Oa  -O  c. 


POWER   FACTOR   OF   THREE-PHASE   CIRCUIT      113 
of  a  single-phase  circuit,  this  expression  may  be  written, 
power/actor^       watts 


volt-amperes ' 

but  in  polyphase  circuits  it  is  not  always  clear  what  is 
to  be  understood  by  the  total  volt-amperes  or  apparent 
power.  In  a  balanced  three-phase  circuit,  with  the  angle 
of  lag,  6,  the  same  on  all  three  phases,  the  true  power  is 
\/3  E  I  cos  0,  and  the  expression  V^  E  I,  which  stands 
for  the  total  volt-amperes,  will  be  more  clearly  recognised 

Tj*  Tj» 

when  put  in  the  form  -\(  —r^  x  I  V  where  -7=-  is  the  star 
J\A/3         /  V3 

voltage  ;  the  total  apparent  power,  in  this  case,  being 
the  sum  of  the  volt-amperes  in  each  branch  of  the  three- 
phase  supply. 

If  the  expressions  for  power  are  divided  by  the 
voltage,  the  power  factor  can  be  expressed  by  the  ratio 
total  energy  current^  ^  ^  h  tfae  most  con 

total  current 

venient  form  of  the  expression  for  general  use,  seeing 
that  the  three  voltages  are  usually  equal  and  symmetrical 
as  regards  phase  angle.  In  the  case  of  a  balanced 
inductive  load,  the  "  total  "  current  for  use  in  the  power 
factor  formula  would  be  3  x  I,  and  the  "  total "  energy 
current  would  be  3  x  I  x  cos  0 ;  but  when  the  load  is 
not  balanced  the  expression  is  not  quite  so  simple.  Not 
only  the  magnitude  of  the  (different)  angles  of  lag,  but 
also  their  sign — i.e.,  whether  the  current  vector  leads  or 
lags  behind  the  star  vector  of  the  E.M.F. — must  be  taken 
into  account ;  this  will  be  explained  in  article  46. 

45.   Vector    Diagram    for    Balanced    Inductive 

Load. — A  condition  very  frequently  met  with  in  practice 
is  that  of  generators  supplying  current  to  induction  motors 

only. 

8 


I  14      MEASUREMENT  AND  CALCULATION  OF  POWER 

In  this  case  the  power  factor  will  be  less  than  unity ; 
but  the  load  will  be  equal  on  all  three  phases. 

Such  a  load  is  shown  diagrammatically  in   Fig.  45. 


FIG.  45. 

Here  the  current  vectors  A,  B,  and  C  are  all  equal  in 
length  ;  but,  instead  of  being  in  phase  with  the  pressure 
vectors  o  a,  o  b,  and  o  c,  they  lag  behind  these  vectors 
by  a  certain  angle  6. 


DIAGRAM    FOR   BALANCED   INDUCTIVE   LOAD      11$ 
The  total  power  is 

W  =  3  (o  a  x  a  i) 

=  3xo0x0A  cos  0 
=  \/3  E  I  cos  B 

where  E  is  the  pressure  between  phases  ; 

I   is  the   amount   of  any  one   of   the  three  line 

currents ;  and 

cos  6  is  the  power  factor  of  the  balanced  three- 
phase  circuit. 

Measurement  of  Power  Factor  on  Balanced  Load. — There 
are  instruments  called  phase  meters,  or  power  factor 
indicators,  so  constructed  as  to  show,  by  the  position  of  a 
pointer,  the  phase  angle  0  between  current  and  E.M.F.  ; 
but,  by  means  of  a  voltmeter,  ammeter,  and  wattmeter, 
this  angle  can  evidently  be  calculated :  the  two  former 
instruments  will  enable  us  to  calculate  the  apparent  power 
(3  x  o  a  x  a  A),  while  the  wattmeter  will  directly  indi- 
cate the  true  power  (3  x  o  a  x  a  i)t  and  the  ratio  which 
this  latter  quantity  bears  to  the  apparent  power  will  be 
the  power  factor  (cos  6)  of  the  three-phase  circuit. 

It  is,  however,  possible  to  determine  the  angle  6,  even 
if  neither  ammeter  nor  voltmeter  is  available,  by  the  use 
of  a  wattmeter  only. 

It  has  already  been  explained  (see  article  42,  p.  106) 
how  the  total  power  of  a  balanced  three-phase  circuit 
may  be  measured  by  means  of  a  single  wattmeter  con- 
nected in  series  with  one  main,  when  the  shunt  coil 
is  connected  alternately  to  each  of  the  other  two 
mains,  the  sum  of  the  two  readings  giving  the  total 
power. 

These  two  wattmeter  readings  will  also  enable  us  to 


Il6      MEASUREMENT  AND  CALCULATION  OF  POWER 

calculate  the  power  factor  of  the  load.  Let  Wx  and  W2 
be  the  two  readings  of  the  wattmeter  ;  then,  if  0  is  the 
angle  of  lag,  it  can  be  shown  that 


tan  e  = 


-  wa) 


W2 


From  this  we  obtain  the  angle  0,  and  cos  0,  the  power 
factor  of  the  balanced  circuit. 

That  the  above  formula  is  correct  for  a  power  factor 
of  unity  is  evident,  because  Wx  will  be  equal  to  W2,  thus 
making  the  numerator  of  the  right-hand  side  of  the 
equation  equal  to  zero  ;  and  if  tan  6  =  o  it  follows  that 
0  =  o,  which  is  the  condition  required  to  make  the 
power  factor  (cos  0)  equal  to  unity.  By  the  aid  of 
Fig.  45  it  will  be  seen  that  the  above  formula  is  correct 
for  any  value  of  the  angle  6. 

Let  us  suppose  the  current  A  to  pass  through  the 
series  coil  of  the  wattmeter,  and  the  reading  Wl  to  be 
taken  with  the  free  end  of  the  shunt  coil  connected 
to  c,  while  for  the  reading  W2  this  connection  is  trans- 
ferred to  b. 

The  power  Wx  will  be  equal  to  the  product  of  the 
volts  c  a  by  the  current  a  A,  or  to  the  length  c  a  multiplied 
by  the  projection  of  a  A  upon  c  a.  Similarly,  the  power 
W2  will  be  equal  to  a  b  multiplied  by  the  projection  of  a  A 
on  a  b. 

It  will  be  observed  that,  in  the  two  readings,  the 
amounts  of  the  main  and  shunt  currents  remain  the  same, 
but  the  phase  of  the  shunt  current  is  different.  The 
results  obtained  by  adding  or  subtracting  the  two 
wattmeter  readings  would  evidently  correspond  with  a 
single  reading  of  the  wattmeter  obtained  by  passing 
through  the  shunt  coil  a  current  equal  to  the  sum  or 
difference  —  as  the  case  may  require  —  of  the  two  separate 


DIAGRAM   FOR   BALANCED   INDUCTIVE   LOAD      117 

shunt  currents.  Now,  the  sum  of  the  two  vectors  c  a  and 
a  b  is  c  b  ;  and  their  difference  —  which  is  the  same  thing  as 
the  sum  of  c  a  with  a  b  reversed  —  is  b  d,  exactly  at  right 
angles  to  b  c,  and  equal  in  length  to  v/^  times  b  c.  We 
may,  therefore,  write 

(  Wl  +  W2)  =  b  d  x  the  projection  of  a  A  upon  b  d 
=  b  d  x  a  i 
=  \/q[  b  c  x  a  i     .....     (i) 

and  (Wl-W2)  =  bcxi  A     ....     (2) 


Inserting  the  values  (i)  and  (2)  in  the  original  equation, 
we  have 


tan  0  = 


x  b  c  x  a  i 


which,  since  a  i  A  is  a  right-angled  triangle,  is  the 
definition  of  the  trigonometrical  tangent  of  the  angle  0, 
and  proves  the  correctness  of  the  formula. 

(A  little  difficulty  may  be  experienced  in  understanding 
why  the  difference  of  the  vectors  c  a  and  a  b  has  been 
taken  to  obtain  the  equivalent  resultant  vector  from 
which  the  sum  of  the  wattmeter  readings  is  calculated, 
and  vice  versa  ;  but  this  is  due  to  the  method  of  changing 
over  the  shunt  connection,  leaving  the  same  end  of  the 
shunt  coil  permanently  connected  to  the  main  A  in  which 
the  wattmeter  is  placed.) 

Example.  —  Let  us  suppose  that  the  wattmeter  reading 
Wj  =  20  kilowatts  and  W2  =  10  kilowatts  ;  then 


20    +    10 

=  '577- 


118      MEASUREMENT  AND  CALCULATION  OF  POWER 

Referring  to  a  table  of  natural  tangents,  we  see  that 
this  corresponds  to  an  angle  0  =  30,  and  since  cos  30  degrees 
=  -866,  this  will  be  the  power  factor  of  the  balanced 
three-phase  circuit, 

46.  Unbalanced  Inductive  Load. — Consider  a  three- 
phase  system  in  which  a  lamp  load  is  taken  off  one  or 
more  phases,  in  addition  to  induction  motors  on  all  three 
phases.  This  may  be  effected  (i)  by  connecting  the 
lamps  in  delta  fashion  directly  between  the  conductors 
A  B,  B  C,  or  C  A,  or  (2)  if  the  generator  armature 
windings  are  star  connected,  by  running  a  fourth  wire,  D, 
back  from  the  load  to  the  neutral  point,  and  connecting 
the  lamps  between  the  terminals  A,  B,  or  C  and  this 
fourth  conductor,  as  shown  jn  Fig.  42  (p.  108).  In  either 
case  the  three  current  vectors  A,  B,  and  C  (Fig.  46)  may 
be  unequal  in  length,  and  make  different  phase  angles, 
a,  /?,  and  y,  with  the  equivalent  star  vectors  drawn  from  the 
point  O  (the  centre  of  the  triangle) ;  but  in  case  (i)  these 
three  vectors  will  form  a  closed  triangle  when  combined 
together,  whereas  in  case  (2)  the  vector  representing  the 
current  (if  any)  in  the  fourth  conductor,  D,  will  be  required 
to  close  the  polygon  of  the  current  vectors. 

The  question  now  arises  as  to  what  is  to  be  understood 
by  the  power  factor  of  such  a  load.  It  may  still  be  defined 
as  the  ratio  of  the  real  power  to  the  apparent  power  ;  but 
although  the  real  power  is  readily  measured  or  calculated 
as  already  explained,  it  is  not  so  easy  to  define  what  is 
now  to  be  understood  by  the  apparent  power  of  the 
unbalanced  three-phase  circuit. 

If,  however,  we  consider  the  total  current  as  being 
made  up  of  two  factors — the  total  "  energy  "  or  "  active  " 
current  and  the  total  "wattless"  or  "reactive"  current — we 
shall,  by  combining  these  at  90  degrees,  obtain  a  quantity 


UNBALANCED   INDUCTIVE   LOAD 


119 


which  may  be  used  for  calculating  the  power  factor  in  all 
cases,  whether  the  load  be  balanced  or  otherwise. 


FIG.  46. 


Thus,  the  "  total "  line  current,  instead  of  being  written 
A  +  B  +  C,  should  be  expressed  as 


V  (a  +  b  +  c)2  +  (ia  +  h  +  ic)*> 


120      MEASUREMENT  AND  CALCULATION  OF  POWER 

where  a,  b,  and  c  stand  for  the  "  energy "  components 
of  the  currents — A  cos  a,  B  cos  /?,  and  C  cos  y — and  ia,  ib> 
and  ic  stand  for  the  "  reactive  "  or  "  wattless  "  components, 
A  sin  a,  B  sin  /3,  and  C  sin  y.  When  the  sign  of  this 
idle  current  component  is  taken  into  account,  it  will  be 
seen  that  the  above  expression  may  give  results  differing 
appreciably  from  the  arithmetical  sum  of  the  idle  currents 
in  the  lines  ;  but,  in  the  event  of  the  load  being  entirely 
non-inductive,  the  algebraic  sum  of  the  idle  currents  will 
be  zero,  thus  making  the  apparent  power,  as  calculated 
by  this  method,  equal  to  the  true  power. 

The  power  factor  of  an  unbalanced  three-phase  load 
may,  therefore,  be  expressed  by  the  ratio 

I™ 


N/   I.2  +    I.2 

where  Iw  =  the  sum  of  all  energy  components  of  current, 
and  !„  =  the  (algebraic)  sum  of  all  wattless  components 
of  current. 

If  we  adopt  the  lettering  of  Fig.  46,  the  power  factor 
would  be  written 

(a  ia  +  b  ib  +  c  ic) 


Example.  —  Suppose  the  pressure  between  terminals  to 
be  500  volts,  and  let  the  phase  angles  be  respectively 

a  =  45  degrees  of  lag, 
ft  =  10  degrees  of  lag, 
7  =  15  degrees  of  lead, 

and  assume  the  main  currents,  as  indicated  by  the  line 
ammeters,  to  be 


UNBALANCED   INDUCTIVE   LOAD  121 

A  =  140  amperes. 
B  =  215 
C  =  105 

Then  a  ia  =  A  cos  a  =     99 

b  ib  =  B  cos  /?  =  2117 
c  ic  =  C  cos  7  =  101-4 

Total     ...    412-1 

Also  A  ia  =  A  sin  a  =  +  99 

B  ib  =  B  sin  /3  =  +  37-3 
C  ic  =  C  sin  7  =  -  27-2 


Total     ...      109-1 
The  real  power  is  therefore 

=j=  x  412-1  =  no  kilowatts, 

and  the  power  factor  is 

412-1  „ 

.7      =  -067.* 

'i)2  +  (109-1)2 

*  If  it  is  desired  to  dispense  with  the  process  of  squaring  and 
extracting  the  square  root,  we  can  use  trigonometrical  tables  thus  : 

Let  cos  9  stand  for  the  power  factor  of  the  three-phase  circuit ; 
then,  since 

mean  sin  6  =  ^  x  109-1 
and  mean  cos  0  =  J  x  412'  i, 

tan  6  =        ,T  =  '2647, 

which  corresponds  to  an  angle  6  of  14°  50'.  By  referring  to  a  table 
of  natural  cosines  we  see  that  cos  14°  50'  =  -967,  which  is  the  same 
result  as  obtained  by  the  more  lengthy  process  of  adding  the 
squares  and  extracting  the  square  root. 

In  the  March  number  of  Power  (1906)  the  author  described  a 


122      MEASUREMENT  AND  CALCULATION  OF  POWER 

47.  Measurement    of   Power    on    High-Tension 

Circuits.  —  In  all  the  foregoing  diagrams,  the  instruments 
— whether  wattmeters  or  ammeters — have  been  shown 
as  being  connected  in  series  with  the  main  leads.  This 
method,  however,  only  applies  to  circuits  of  pressures 
not  exceeding,  say,  500  volts.  For  higher  pressures, 
transformers  are  generally  used,  in  order  that  low- 
tension  connections  only  may  be  brought  to  the  terminals 
of  the  instruments.  Even  on  low-tension  systems,  series 


Wattmeter 


FIG,  47. 

transformers  would  be  used  to  reduce  a  large  current  to 
a  convenient  amount,  such  as  3  or  5  amperes,  in  the 
instruments.  The  various  methods  of  power  measure- 
ments are  not  altered  thereby  ;  it  is  merely  necessary  to 
consider  the  transformers  as  suitable  pieces  of  apparatus 


simple  graphical  method  which  requires  only  four  measurements  to 
be  taken  off  the  diagram,  and  yet  gives  all  particulars  for  calculating 
the  "  real  "  and  the  "  idle  "  power,  and  hence  the  power  factor  of 
any  three-phase  circuit  whether  balanced  or  not. 


MEASUREMENT   OF   POWER   ON    H.T.   CIRCUITS       123 

for  providing  secondary  pressures  or  currents — as  the 
case  may  be — exactly  proportional  to,  and  of  the  same  phase 
as,  the  primary  pressures  or  currents. 

As  an  example,  the  diagram  Fig.  47  has  been  drawn. 
It  illustrates  a  suitable  arrangement  for  measuring  the 
total  power  of  a  balanced  high-tension  three-phase  circuit. 

Here  A  and  a  are  the  primary  and  secondary  wind- 


i      ...A 


FIG.  48. 

ings  of  the  current  transformer  connected  in  series  with 
one  of  the  mains.  This  sends  a  current  through  the 
main  coil,  M,  of  the  wattmeter,  of  the  same  phase  as  the 
current  A,  the  actual  amount  of  which  will  depend  upon 
the  ratio  of  the  primary  and  secondary  turns,  but  which 
— in  any  case — will  be  proportional  to  A. 
The  two  pressure  transformers,  each  with  T  turns  in  the 


124      MEASUREMENT  AND  CALCULATION  OF  POWER 

primary  winding  and  t  turns  in  the  secondary,  have 
their  primaries  connected  up  between  A  and  B,  and 
A  and  C,  respectively,  while  the  secondary  windings  are 
connected  in  series,  and  provide  the  shunt  current  for  the 
pressure  coil,  Z,  of  the  wattmeter. 

Fig.  48  will  serve  to  explain  how  the  wattmeter  can 
be  made  to  indicate  correctly  the  total  output  of  the 
balanced  circuit. 

It  will  be  seen  that  the  secondary  winding  of  one 
pressure  transformer  (Fig.  47)  is  reversed  before  connect- 
ing in  series  with  the  other  transformer  and  the  pressure 
coil  of  the  wattmeter.  This  means  that  the  resultant 
current  through  the  pressure  coil  will  not  be  propor- 
tional to  the  sum  of  the  vectors  b  a  and  a  c  (Fig.  48),  but 
to  their  difference,  which  is  represented  by  the  dotted 
vector  b  v.  This  last  is  at  right  angles  to  c  b,  and  parallel 
to  O  a\  and,  since  the  total  output  is  equal  to  three 
times  O  a  x  A  cos  6,  or  to 

3  (O  a)  x  (a  i), 

it  is  evidently  only  a  question  of  calibration  and  of  the 
ratios  of  turns  in  the  various  transformer  windings,  to 
insure  that  the  single  wattmeter  shall  register  the  exact 
total  output  of  the  three-phase  high-tension  balanced 
circuit. 


CHAPTER  V 

POLYPHASE    TRANSFORMERS 

48.  Theory  of  the  Single-Phase  Transformer. — 

A  single-phase  alternating-current  transformer  may  be 
considered  as  consisting  of  a  core  of  laminated  iron  upon 
which  are  wound  two  sets  of  coils,  known  as  the  primary 
and  secondary  windings  respectively. 

If  an  alternating  E.M.F.  is  applied  to  the  terminals  of 
the  primary,  this  will  lead  to  a  certain  flux  of  alternating 
magnetism  being  set  up  in  the  iron  core,  which,  in  its 
turn,  will  induce  a  counter  E.M.F.  of  self-induction  in 
the  primary  winding,  the  action  being  that  of  a  choking 
coil  (see  article  19,  Chapter  II.).  But  since  the  secondary 
circuit — although  not  in  electrical  connection  with  the 
primary — is  wound  on  the  same  iron  core,  the  variations 
of  magnetic  flux  which  induce  the  back  E.M.F.  in  the 
primary  will,  at  the  same  time,  generate  an  E.M.F.  in 
the  secondary  coils. 

The  path  of  the  magnetic  lines  is  usually  through 
a  closed  iron  circuit ;  and,  although  in  practice  there  is 
always  a  certain  amount  of  leakage  or  stray  magnetism 
which  is  not  enclosed  by  the  secondary  windings,  the 
effects  of  this  magnetic  leakage  are  very  small  in  all 
well-designed  transformers,  and  it  will  somewhat  sim- 
plify the  diagrams  if  we  neglect  this  entirely.  The 
assumption  is,  therefore,  that  the  whole  of  the  magnetism 

125 


126        POLYPHASE  TRANSFORMERS 

required  to  produce  the  necessary  back  E.M.F.  in  the 
primary  coil  passes  also  through  the  secondary  coils  — 
that  is  to  say,  the  induced  E.M.F.  per  turn  of  wire  is 
supposed  to  be  exactly  the  same  in  the  secondary  as  in 
the  primary  coil. 

Suppose  now  that  the  two  ends  of  the  primary  winding 
are  connected  to  constant-pressure  mains,  and  that  no 
current  is  taken  from  the  secondary  winding.  Under 
these  conditions,  the  primary  circuit  acts  simply  as  a 
choking  coil,  of  which  the  self-induction  is  so  great,  and 
the  ohmic  resistance  relatively  so  small,  that  no  current 
passes,  except  the  very  small  amount  required  to  mag- 
netise the  core.  The  induced  E.M.F.  is,  therefore, 
practically  equal  and  opposite  to  the  applied  potential 
difference  at  primary  terminals,  and  the  relation  between 
the  magnetic  flux  in  the  core  and  the  primary  impressed 
E.M.F.  will  be  given  by  the  equation 


io8     ' 

where  E>m  stands  for  the  mean  value,  in  volts,  of  the 
primary  E.M.F.,  and  n  is  the  frequency. 

The  number  of  turns,  S,  in  the  primary   of   a  well- 

designed   transformer   is   always   such  that  the  current 

required  to  produce  the  magnetic  flux  N  is  very  small  ; 

it  is  generally  somewhere  between  2  per  cent,  and  5  per 

.  cent,  of  the  full-load  current. 

Although  the  rise  and  fall  of  the  magnetism  will  be 
a  quarter  of  a  period  out  of  phase  with  the  E.M.F.,  the 
open-circuit  primary  current  will  not  be  entirely  "  watt- 
less," but  may  be  considered  as  made  up  of  two  com- 
ponents —  the  "  wattless  "  or  true  magnetising  com- 
ponent, in  phase  with  the  magnetism,  and  the  "  energy  " 
component,  due  to  hysteresis  and  eddy  currents,  in  phase 


THEORY  OF   SINGLE-PHASE   TRANSFORMER 

with  the  impressed  E.M.F.  The  reader  is,  however, 
referred  to  article  24  (p.  50),  where  the  question  of  mag- 
netising current  for  a  circuit  containing  iron  has  already 
been  dealt  with. 

Since  the  secondary  and  primary  coils  are  both  wound 
on  the  same  core,  it  follows  that  the  actual  volts  induced 
will  be  directly  proportional  to  the  number  of  turns  of 
wire  in  either  coil. 

Thus,  if  the  primary  winding  consists  of  1,000  turns  of 
wire,  all  in  series,  while  the  secondary  has  only  50  turns, 
the  ratio  of  turns  is  20  :  i,  and  this  will  also  be  the 
ratio  of  primary  impressed  E.M.F.  to  secondary  induced 
E.M.F.  on  the  assumption  of  there  being  no  magnetic 
leakage  and  a  negligible  I R  drop  in  primary. 

For  convenience  in  drawing  the  diagrams,  we  shall,  in 
all  cases,  suppose  the  primary  and  secondary  windings  to 
have  the  same  number  of  turns ;  the  transforming  ratio 
will  therefore  be  i  :  i,  and  the  pressure  obtained  at 
secondary  terminals  will  be  exactly  equal,  but  opposite  in 
phase,  to  the  applied  primary  pressure. 

This  last  statement  will  only  be  strictly  correct  if  we 
also  assume  the  ohmic  resistance  of  the  coils  to  be  negli- 
gible ;  but  the  pressure  drop  due  to  this  internal  resistance 
is  always  small,  and  rarely  exceeds  2  per  cent.,  even  in 
small  transformers.  It  is  easily  taken  into  account,  if 
desired ;  but  since  the  principles  of  action  only  will 
receive  our  attention,  details  of  design  will  not  be  dealt 
with,  and  we  may,  therefore,  neglect  the  ohmic  resistance 
of  both  primary  and  secondary  coils.  Imagine,  now,  that 
the  secondary  circuit  of  a  transformer,  with  primary  on 
constant-pressure  mains,  is  closed  through  a  resistance ; 
the  resulting  current  will  produce  a  magnetising  force  in 
the  core.  This  magnetising  force  will  not  produce  a 
change  in  the  magnetism,  because  it  will  be  instantly 


128 


POLYPHASP:  TRANSFORMERS 


counteracted  by  a  change  in  the  primary  current,  which 
will  so  adjust  itself  as  to  maintain  the  same  (or  nearly 
the  same)  cycle  of  magnetisation  as  before — that  is  to 
say,  the  flux  will  continue  to  be  such  as  will  induce  an 
E.M.F.  in  the  primary  windings  equal  but  opposite  to 
the  primary  impressed  potential  difference.  These  two 
opposing  pressures  can  evidently  not  be  exactly  equal,  or 
no  current  would  flow  in  the  primary  coils ;  but  since,  in 


FIG.  49. 

practice,  the  primary  resistance — although  not  of  zero 
value — is  relatively  small,  it  will  be  understood  that 
a  very  small  resultant  pressure  across  the  primary  ter- 
minals will  cause  a  very  large  current  to  flow  through  the 
coils. 

In  Fig.  49,  E!  is  the  curve  of  primary  impressed 
E.M.F.,  and  ll  is  the  magnetising  current,  distorted  by 
the  hysteresis  of  the  iron  core.  E2  is  the  curve  of 
secondary  E.M.F.,  which  coincides  in  phase  with  the 


THEORY   OF   SINGLE-PHASE   TRANSFORMER        12Q 

primary  induced  E.M.F.,  and  is  therefore — on  account 
of  the  comparatively  small  ohmic  resistance  of  the 
primary — almost  exactly  in  opposition  to  the  impressed 
E.M.F.  The  curve  of  magnetisation  (not  shown)  would 
be  exactly  a  quarter  period  in  advance  of  the  secondary 
E.M.F. 

In  Fig.  50  the  secondary  circuit  is  closed  through  its 
proper  load  of  incandescent  lamps.     There  is  no  appreci- 


FIG.  50. 


able  self-induction  in  such  a  circuit,  and  the  secondary 
current  will,  therefore,  be  in  step  with  the  secondary 
E.M.F.  For  simplicity  it  is  represented  in  Fig.  50  by 
the  same  curve  as  E2. 

The  tendency  of  this  secondary  current  being  to  weaken 
the  magnetism  in  the  core,  and  therefore  diminish  the 
primary  induced  E.M.F.,  it  follows  that  the  current  in 
the  primary  will  grow  until  the  magnetism  is  again  of 
such  an  amount  as  to  restore  balance  in  the  primary 

9 


130 


POLYPHASE   TRANSFORMERS 


circuit.  Hence,  neglecting  the  small  loss  of  volts  due 
to  the  increased  primary  current,  the  induction  through 
the  primary  must  remain  as  before ;  and  the  new  current 
curve,  Ij  (Fig.  50),  is  obtained  by  adding  the  ordinates  of 
the  current  curve  in  Fig.  49  to  those  of  another  curve, 
exactly  opposite  in  phase  to  the  secondary  current,  and 
of  such  a  value  as  to  produce  an  equal  magnetising  effect. 

49.  Vector    Diagram   of    Transformer   without 
Leakage. — In  Fig.  51  let  O  E2  represent  the  secondary 


FIG.  51. 

E.M.F.  Then,  on  the  assumption  that  the  voltage  drop 
in  primary,  due  to  ohmic  resistance,  is  negligible — which 
assumption  is  generally  permissible— the  primary  im- 
pressed E.M.F.,  E,  will  be  exactly  opposite  in  phase  to 
the  secondary  or  induced  E.M.F.,  and  O  E  will  be  equal 
to  O  E2  multiplied  by  the  ratio  of  turns  in  the  two 
windings.  But,  since  we  have  assumed — for  uniformity 
and  convenience  of  construction — that  the  transforming 
ratio  is  i  :  i,  the  length  O  E  must  be  made  equal  to 
OE0. 


DIAGRAM  OF  TRANSFORMER  WITHOUT  LEAKAGE    131 

With  regard  to  the  magnetising  current  in  the  primary 
coil,  which  will  be  very  small,  this  will  consist  of  the 
"  wattless  "  or  true  exciting  current  O  10,  in  phase  with 
the  induction — and,  therefore,  90  degrees  in  advance  of 
O  E2 — and  the  "  active  "  component,  O  Iw,  in  phase 
with  O  E.  This  "  active "  component  is  due  partly  to 
hysteresis  and  partly  to  eddy  currents,  as  was  explained 
in  Chapter  II.,  article  24,  and  illustrated  by  Fig.  20. 

The  total  magnetising  current,  O  Im,  may  now  be 
drawn;  it  will  lag  behind  the  impressed  E.M.F.  by 


FIG.  52. 

about  45  degrees  in  a  well-designed  transformer,  which 
corresponds  to  a  power  factor,  on  open  secondary  circuit, 
of  about  7. 

Effect  of  Closing  Secondary  on  Non-Inductive  Load. — The 
load  being  non-inductive,  the  secondary  current,  O  I2 
(Fig.  51),  will  be  in  phase  with  the  secondary  E.M.F. 
It  will  be  balanced  by  a  component,  O  Ilf  of  the  primary 
current,  exactly  equal  and  opposite  to  O  I2  (and  there- 
fore in  phase  with  E);^and  the  total  primary  current 
will  now  be  represented  by  the  resultant  O  I. 


132        POLYPHASE  TRANSFORMERS 

Effect  of  Closing  Secondary  on  Partly  Inductive  Load. — 
In  Fig.  52  let  O  E2  be  the  secondary  E.M.F.  as  before, 
and  O  I2  the  secondary  current,  which  now  lags  some- 
what behind  this  E.M.F.  The  balancing  component  of 
the  primary  current  will  still  be  equal  and  opposite  to 
O  I2,  with  the  result  that  the  primary  current,  O  I,  will 
also  lag  behind  the  impressed  E.M.F.  It  will  be  evident 
from  inspection  of  the  diagram  that  the  energy  put  into 
the  primary  is  still  in  excess  of  the  energy  taken  out 
at  the  secondary  terminals  by  the  amount  lost  in  hyster- 
esis and  eddy  currents  in  the  core. 

50.  Polyphase  Transformers. — On  any  polyphase 
system,  it  is  always  possible  to  use  two  or  more  single- 
phase  transformers,  suitably  connected  up,  for  the  purpose 
of  raising  or  lowering  the  pressure  of  the  supply  leads. 

The  use  of  single-phase  transformers  has  much  to 
recommend  it,  as,  in  the  event  of  a  breakdown,  the  repairs 
are  generally  more  quickly  and  more  economically  carried 
out. 

Indeed,  for  large  outputs,  it  is  customary  to  use  separate 
single-phase  transformers  on  the  various  phases.  A  poly- 
phase transformer,  built  up  with  a  common  iron  core, 
might  become  very  large ;  thus,  on  a  three-phase  circuit, 
it  would  be  required  to  deal  with  three  times  the  output 
of  each  single-phase  transformer,  and,  apart  from  the 
difficulties  of  manufacture  and  handling,  there  would  be 
more  difficulty  in  providing  adequate  cooling  surface,  with 
the  result  that  the  saving  in  cost  of  material  required  would 
be  very  small. 

In  the  case  of  transformers  for  small  outputs  it  is, 
however,  cheaper  to  arrange  the  windings  on  a  common 
iron  core.  Figs.  53  and  54  show  sections  through  a  two- 
phase  and  three-phase  transformer  respectively. 

In  the  former,  the  primary  and  secondary  coils  belong- 


POLYPHASE   TRANSFORMERS 


133 


ing  to  phase  A  are  wound  on  the  left-hand  limb  of  the 
closed  iron  circuit,  while  both  coils  of  phase  B  are  wound 
on  the  right-hand  limb ;  these  two  limbs  being  of  equal 
section.  In  the  centre,  between  the  two  bobbins  or  sets 
of  coils,  a  common  path  for  the  return  magnetism  is  pro- 
vided, and  since  this  will  have  to  carry  either  the  sum  or 
the  difference  of  the  two  equal  alternating  fields  having 
a  phase  difference  of  90  degrees,  the  centre  core — if 
designed  for  the  same  flux  density  as  the  two  outer 
cores — must  have  a  cross-section  Va  times  as  great  as 
either  A  or  B. 


B 


FIG.  53. 

In  Fig.  54  the  three  iron  cores  are  of  equal  section, 
and  they  are  each  wound  with  the  primary  and  secondary 
coils  belonging  to  one  of  the  three  phases.  With  this 
arrangement  in  place  of  three  separate  transformers,  if 
the  flux  density  is  supposed  to  be  the  same  in  each  case, 
the  saving  of  iron  in  the  magnetic  circuit  will  be  seen  to 
be  quite  appreciable :  it  will  be  noticed  that  each  of  the 
three  iron  cores  forms  the  return  path  for  the  magnetic 
flux  in  the  other  two,  in  the  same  manner  as  the  three 
conductors  of  a  balanced  three-phase  circuit  suffice  to 


134 


POLYPHASE   TRANSFORMERS 


carry  the  current  to  and  from  the  apparatus  constituting 
the  load. 

5 1 .  Methods  of  Connecting  Three-Phase  Trans- 
formers.— If  it  is  desirable  or  necessary  to  have  the 
neutral  point  available,  the  secondary  windings  of  the 
three-phase  transformers  must  be  Y-connected ;  and  for 
small  transformers — in  which  the  space  occupied  by 
insulation  is  large  in  proportion  to  the  cross-section  of 
the  copper  in  the  windings — a  saving  in  cost  and  weight 
is  effected  by  adopting  this  mode  of  connection. 


FIG.  54. 

The  chief  advantage  of  connecting  up  the  windings 
in  A  fashion,  more  especially  when  three  separate  trans- 
formers are  used,  is  that,  in  the  event  of  one  of  the 
transformers  in  a  group  of  three  breaking  down,  it  can 
be  entirely  cut  out  of  circuit,  leaving  the  two  remaining 
sound  transformers  to  provide  the  necessary  three-phase 
pressure;  but  this  will  not  be  possible  if  the  windings 
are  star-connected,  since,  with  this  arrangement,  any  two 
windings  taken  together  will  only  provide  the  pressure 
across  one  phase. 

With  the  A  connection,  the  removal  of  one  side  of  the 


CONNECTING  THREE-PHASE   TRANSFORMERS      135 

mesh  virtually  converts  the  arrangement  into  a  two- 
phase  system  having  a  common  return  carrying  the 
same  amount  of  current  as  each  of  the  other  two  con- 
ductors, instead  of  carrying  a  current  v2  times  greater, 
as  would  be  the  case  with  the  more  usual  two-phase 
system  in  which  the  phase  angle  is  90  degrees.  This  is 


FIG.  55. 

known  as  the  open  delta  connection ;  but  it  is  used  only 
in  cases  of  emergency. 

It  is  not  necessary  that  the  primary  windings  be  con- 
nected up  in  a  similar  manner  to  the  secondary  windings, 
but  if  the  primary  is  star-connected  and  the  secondary 
A-connected,  or  vice  versa,  the  ratios  of  the  number  of 
turns  in  the  primary  and  secondary  coils  will  no  longer 
correspond  with  the  transforming  ratio  as  measured  across 
the  phases.  The  diagrams  in  Fig.  55  will  make  this 


136        POLYPHASE  TRANSFORMERS 

clear.  Here  the  upper  system  of  vectors  indicates  the 
amount  and  direction  of  the  pressures  in  the  primary 
coils,  while  the  lower  set  of  vectors  indicates  the  pressures 
on  the  secondary  side. 

The  left-hand  combination  refers  to  a  star-connected 
primary  and  a  delta-connected  secondary,  the  ratio  of 

transformation  being -/=-,  or -577  times  the  ratio  of  turns; 

while  in  the  right-hand  combination  we  have  a  A-con- 
nected  primary  and  a  Y-connected  secondary ;  the  trans- 
forming ratio  in  this  case  being  ^3,  or  1732  times  the 
ratio  of  secondary  to  primary  turns. 

It  is  interesting  to  note  that,  in  both  these  combinations 
of  windings,  the  secondary  pressures  as  measured  between 
the  three  terminals  are  not  180  degrees  out  of  phase  with 
the  primary  pressures  (as  would  be  the  case  if  the  wind- 
ings were  connected  up  similarly  on  both  high-tension 
and  low-tension  sides),  but  differ  in  phase  by  a  right 
angle,  or  90  degrees. 

52.  Efficiency  of  Transformers. — The  efficiency  of 
the  alternating-current  transformer  is  very  high ;  its  actual 
value  will  depend,  to  a  certain  extent,  upon  the  skill  and 
knowledge  of  the  designer,  but  more  especially  upon  the 
amount  and  quality  of  the  materials  used  in  its  construc- 
tion. 

The  efficiency  should  never  be  considered  apart  from 
first  cost,  or  without  reference  to  the  efficiency  as  a  whole 
of  the  system  in  connection  with  which  the  transformers 
are  to  be  used. 

It  must  also  be  borne  in  mind  that  the  temperature 
rise  is  no  indication  of  the  amount  of  power  lost  in  the 
transformer ;  a  transformer  which  gets  very  hot  may  be 
more  efficient  than  another  which  remains  comparatively 


EFFICIENCY   OF   TRANSFORMERS  137 

cool.  Temperature  rise  must  only  be  considered  with 
reference  to  the  effect  it  may  have  upon  the  materials 
used  in  the  construction  of  the  transformer ;  it  is  a  ques- 
tion which  concerns  the  manufacturer. 

The  efficiency  of  a  small  transformer  will  necessarily 
be  lower  than  that  of  a  larger  transformer.  The  maximum 
efficiency  will  generally  be  reached  at  from  three-quarters 
to  full  load,  and  it  is  important  that  all  transformers  on 
a  given  system  be  arranged  so  that  the  hours  during 
which  they  supply  a  light  load  may  be  as  few  as 
possible. 

As  approximate  figures  for  well-designed  transformers, 
whether  single-  or  poly-phase,  -95  to  -96  may  be  taken  as 
the  full-load  efficiency  of  a  3-kw.  transformer,  while  if  the 
output  is  one  hundred  times  as  great  (say  300  kw.),  the 
efficiency  might  be  as  high  as  98-5  per  cent. 

53.  Phase  Transformation. — In  the  last  chapter 
reference  was  made  to  the  principle  of  replacing  the 
usual  three-phase  triangle  of  pressure  vectors  by  any 
three  star  vectors  radiating  from  a  common  point  O, 
and  terminating  at  the  vertices  of  the  triangle. 

Such  a  system  of  vectors  will  produce  exactly  the 
same  potential  differences  between  the  three  terminals 
as  the  original  mesh  connection  which  it  is  designed  to 
replace. 

Scott's  System. — The  above  principle  has  been  made 
use  of  by  Mr.  C.  F.  Scott  in  his  ingenious  arrangement 
for  changing  from  two  to  three  phases  by  means  of  static 
transformers  only. 

In  Fig.  56  the  left-hand  diagram  shows  the  usual 
three-phase  triangle  of  vectors,  which  may  be  replaced 
by  the  three  vectors  O  A,  O  B,  and  O  C,  all  radiating 
from  the  common  point  O.  This  point  lies  on  the  centre 


138 


POLYPHASE   TRANSFORMERS 


of  the  line  A  C,  and  the  line  O  B  is,  therefore,  at  right 
angles  to  A  C.  In  this  manner  the  three  vectors  A  B, 
B  C,  and  C  A  can  be  replaced  by  two  vectors  O  A  and 
O  C  exactly  equal  and  opposite,  and  a  third  vector  O  B 
having  a  phase  difference  of  90  degrees  with  either  of 
the  other  two. 

The  right-hand  diagram  in  Fig.  56  shows  the  trans- 
former connections. 


FIG.  56. 


Two  transformers  are  needed,  with  a  connection  from 
the  end  of  one  secondary  to  the  centre  point  of  the  other 
secondary. 

If  the  primaries  of  the  transformers,  T1  and  T2,  are 
fed  respectively  by  phase  I.  and  phase  II.  of  a  two-phase 
supply  having  a  phase  angle  of  90  degrees,  the  pressures 
generated  in  the  transformer  secondaries,  St  and  S2,  will 
also  be  90  degrees  out  of  phase.  It  is,  therefore,  merely 


SCOTT'S  SYSTEM   OF   PHASE   TRANSFORMATION      139 

necessary  so  to  proportion  the  number  of  turns  in  the 
windings  that  the  combination  of  these  E.M.F.s  will 
produce  equal  pressures  between  the  pairs  of  terminals 
A  B,  B  C,  and  C  A.  If  the  transforming  ratio  of  trans- 
former T2  is  i  :  i,  the  pressure  measured  across  A  C  will 
be  the  same  as  the  two-phase  supply  pressure ;  and  if  the 
ratio  of  turns  in  transformer  Tl  were  also  i  :  i,  the 
pressure  O  B  (see  left-hand  diagram,  Fig.  56)  would  be 
equal  in  amount  to  A  C.  This  would  not  be  correct : 
the  length  of  the  vector  O  B  is 

O  B  =  B  C  sin  60  degrees 

=  A  C  sin  60  degrees 
=  A  C  x  -866, 

and  it  follows  that,  if  the  ratio  of  primary  to  secondary 
turns  in  T2  is  i  :  i,  the  ratio  of  turns  in  Tl  must  be 
i  :  '866  in  order  that  the  triangle  ABC  may  be  equi- 
lateral. 

Example. — Let  us  assume  the  two-phase  pressure  to 
be  2,000  volts,  which  it  is  desired  to  transform  into  three- 
phase  at  a  pressure  of  30,000  volts. 

Suppose  the  primary  turns  on  transformers  T1  and  T2 
are  200  in  each  case  ;  then,  on  each  half  of  the  secondary 
winding  S2,  the  required  number  of  turns  is 

S2  _  i        200  x  30,000 
2       2  2,000 

=  1,5°°; 

while  the  secondary  turns  on  Tl  must  be 

200  x  30,000 
^  -  -  ^oocT       *  '866 

=  2,600. 


140 


POLYPHASE   TRANSFORMERS 


Lunfs  System. — The  system  of  phase  transformation 
described  above  is  reversible — that  is  to  say,  it  can  be 
used,  if  desired,  for  changing  from  three  to  two  phase. 

Another  method  of  effecting  this  latter  transformation 
is  due  to  Mr.  A.  D.  Lunt.  The  connections  of  the  two 
transformers  are  shown  in  the  left-hand  diagram  of 
Fig.  57,  while  the  right-hand  diagram  shows  the  vectors 
of  the  various  magnetic  fluxes  in  the  cores  of  the  trans- 
formers. 


TMTHI 


FIG.  57. 

Considering  first  the  left-hand  transformer,  there  will 
be  an  equal  number  of  turns  in  the  windings  A  and  B, 
producing  equal  magnetic  fluxes,  O  A  and  O  B,  differing 
in  phase  by  120  degrees.  The  resultant  flux  through  the 
central  core  will  be  O  C. 

With  regard  to  the  right-hand  transformer,  the  number 
of  turns  in  the  coil  D  will  be  smaller,  and  in  the  coil  E 
greater,  than  in  coils  A  or  B,  the  actual  number  of  turns 
being  such  as  to  produce  magnetic  fluxes  in  the  cores 
D  and  E,  of  the  second  transformer,  proportional  to  the 
lengths  of  the  vectors  O  D  and  O  E.  These  vectors  must 


LUNT'S   SYSTEM   OF    PHASE   TRANSFORMATION       14! 

necessarily  subtend  a  phase  angle  of  120  degrees;  but, 
owing  to  their  lengths  being  suitably  proportioned,  they 
will  produce  a  resultant  flux,  O  F,  through  the  central 
core,  F,  exactly  at  right  angles  to  the  flux  O  C,  in  the 
core  C  of  the  first  transformer. 

It  follows  that  a  two-phase  supply  can  be  taken  from 
secondary  coils,  having  the  same  number  of  turns,  wound 
on  the  central  cores  of  the  two  transformers. 


CHAPTER  VI 

POWER    TRANSMISSION    BY    POLYPHASE    CURRENTS 

54.  ONE  of  the  chief  reasons  why  polyphase  currents 
are  used  almost  to  the  exclusion  of  other  systems  of 
power  transmission  and  distribution  lies  in  the  ease  with 
which  such  currents  can  be  transmitted  over  great 
distances  at  high  pressures  to  the  centres  of  population, 
and  there  converted  by  means  of  static  transformers  to 
pressures  suitable  for  distribution.  When  the  frequency 
is  not  too  low — not  less  than  forty  cycles  per  second — both 
induction  motors  and  incandescent  lamps  can  be  connected 
to  the  same  secondary  circuits.  If  preferred,  the  three- 
phase  currents,  transmitted  from  a  water-power  site,  or 
from  a  coal-mining  district,  can  be  used  to  drive  polyphase 
motors  at  the  receiving  sub-stations ;  and  these  motors 
can,  in  their  turn,  be  coupled  to  any  kind  of  machine  or 
generator,  such  as  D.C.  dynamo  machines,  if  continuous 
currents  are  required. 

It  is  not  proposed,  in  this  chapter,  to  consider  in  detail 
the  many  problems  arising  in  connection  with  the  trans- 
mission of  energy  by  polyphase  currents,  not  only  because 
the  writer  has  dealt  with  this  subject  elsewhere,*  but 
mainly  because  the  scope  of  this  book  does  not  permit 
of  any  question  of  design  or  of  practical  details  being 
referred  to  otherwise  than  superficially.  An  attempt 

*  "Overhead  Electric  Power  Transmission"  (McGraw-Hill 
Book  Company). 

142 


LOSSES   IN    TRANSMISSION  143 

will,  however,  be  made  to  put  before  the  reader  the 
essential  principles  with  which  the  electrical  engineer  is 
concerned,  involving  of  necessity  some  economic  con- 
siderations ;  but  the  mechanical  problems,  involving 
strengths  of  poles  and  wires,  will  not  be  touched  upon. 

Transmission  by  single-,  two-,  and  three-phase  currents 
will  be  considered  in  succession,  the  greatest  amount  of 
space  being  devoted  to  single-phase  currents,  for  the 
reason  that  all  polyphase  transmissions  can  be  treated  as 
a  combination  of  several  single-phase  transmissions ;  and 
a  proper  understanding  of  the  conditions  met  with  in 
transmitting  single-phase  energy  over  two  wires  is 
therefore  essential  for  the  solution  of  problems  in  the 
transmission  of  energy  by  polyphase  currents. 

55.  Losses  in  Transmission. — With  high  voltages, 
such  as  are  necessary  for  the  economical  transmission  of 
electric  power  to  a  distance,  we  have  to  consider,  not 
only  the  losses  due  to  the  heating  of  the  conductors  by 
the  current,  but  also  the  power  lost  in  the  dielectric 
forming  the  insulation  of  the  cables,  or,  in  the  more  usual 
case  of  overhead  lines,  the  discharge  at  the  surface  of  the 
conductors,  and  the  waste  of  power  in  the  air.  This  loss 
of  power  accompanies  the  visible  halo  of  light  known  as 
the  "  corona."  It  is  inappreciable  except  at  the  higher 
voltages ;  and  when  the  pressure  of  transmission  is  below 
60,000  volts,  it  is  rarely  necessary  to  take  into  account 
the  possible  increase  of  power  loss  due  to  corona 
formation. 

The  actual  amount  of  the  losses  will  depend  upon  the 
ratio  of  the  spacing  between  conductors  to  the  diameter 
of  the  wires.  It  will  be  directly  proportional  to  the 
frequency  and  length  of  line,  and  will  increase  as  the 
square  of  the  amount  by  which  the  actual  voltage  exceeds 
a  certain  value  known  as  the  disruptive  critical  voltage, 


144  POWER   TRANSMISSION 

this  last  being  practically  constant  for  any  given  diameter 
and  spacing  of  wires. 

In  addition  to  these  two  causes  of  power  dissipation, 
we  must  not  overlook  the  leakage  over  insulators  on  a 
long  high-tension  aerial  line ;  but  the  amount  of  current 
which  leaks  to  earth,  or  between  conductors,  in  this 
manner  is  very  small  on  a  carefully-designed  and  well- 
constructed  line,  and  it  is  generally  permissible  to  neglect 
it.  The  allowable  loss  of  power  in  a  transmission  line  is 
a  matter  of  the  greatest  importance,  and  it  will  be  again 
referred  to  after  we  have  considered  the  various  systems 
and  arrangements  of  conductors;  but  for  the  present 
a  certain  percentage  loss  in  transmission  will  be  assumed, 
apart  from  economical  considerations. 

56.  Choice  of  Voltage.  —  Generally  speaking,  the 
higher  the  pressure  the  more  economical  will  be  the 
transmission,  because,  for  a  given  total  power,  the 
current,  and,  therefore,  the  weight  of  the  conductors,  is 
reduced.  But  there  are,  obviously,  many  considerations 
which  stand  in  the  way  of  very  high  pressures  being  used 
in  all  cases  ;  and  each  particular  scheme  must  be  examined 
from  every  possible  point  of  view  before  definitely  deciding 
upon  the  voltage  to  be  adopted.  In  the  first  place,  a 
small  current  at  a  very  high  pressure  costs  more  to  pro- 
duce than  a  larger  current  at  a  proportionally  lower 
pressure,  either  because  the  generators  themselves  will 
be  more  costly,  or  because  step- up  transformers  will  have 
to  be  used.  If,  however,  the  line  losses  would  be  very 
considerable  with  the  larger  currents,  then  it  might  be 
more  economical  to  lay  out  capital  in  step-up  trans- 
formers, extra  insulation  of  conductors,  and  step-down 
transformers  at  the  receiving  end,  and  so  save  copper  in 
the  lines.  In  other  words,  the  most  suitable  voltage  will 
depend  very  .largely  upon  the  distance  to  which  the  energy 


CHOICE   OF   VOLTAGE  145 

has  to  be  transmitted,  but  also  upon  the  total  amount  of 
this  energy. 

The  determination  of  the  proper  voltage  to  be  used  on 
any  given  transmission  scheme  is  mainly  an  economic 
question  ;  but,  as  a  rough  approximation  for  preliminary 
calculations,  the  following  empirical  formula  may  be 
used : 

Pressure  between  wires, \  /r»  .  K.W. 

in  kilo-volts  /  =     5'5  A/  J       -750" 

— where  D  =  distance  of  transmission  in  miles,  and 
K.W.  =  total  kilowatts  transmitted. 

There  are  overhead  transmission  lines  actually  operating 
at  pressures  up  to  150,000  volts,  and  it  is  probable  that 
the  near  future  will  see  energy  transmitted  at  pressures 
in  the  neighbourhood  of  200,000  volts ;  but  the  higher 
pressures  should  never  be  adopted  without  due  regard  to 
economical  principles. 

With  reference  to  underground  cables,  these  are  quite 
out  of  the  question  in  connection  with  long-distance 
transmission  by  alternating  currents.  Their  cost  would 
be  prohibitive,  and,  apart  from  this,  they  are  unsuitable 
for  very  high  pressures.  It  is  unlikely  that  insulated 
cables  will  be  used  in  the  near  future  for  pressures  much 
in  excess  of  50,000  volts,  not  only  because  of  their  exces- 
sive cost,  but  also  on  account  of  the  very  considerable 
capacity  effects  and  large  dielectric  losses.  There  will, 
however,  always  be  a  use  for  insulated  cables  at  compara- 
tively high  pressures,  in  connection  with  transmission 
schemes,  because  of  the  objections  to  carrying  the  over- 
head conductors,  at  extra  high  pressures,  through  towns 
or  populous  districts. 

57.  Transmission  by  Single-Phase  Alternating 
Currents. — Let  us  assume,  in  the  first  place,  that 

10 


146  POWER   TRANSMISSION 

not  only  the  load  at  the  distant  end,  but  also  the  trans- 
mission line,  are  without  inductance  or  capacity. 

If  E  =  volts  between  conductors  at  generating  end ; 
I  =  the  current  in  amperes ; 
R  =  resistance  of  "  load  "  at  distant  end  ; 
v  =  resistance  of  each  of  the  two  conductors ; 

then  the  total  power  transmitted  will  be 

W  =  E  x  I 

=  I2  x  (R  +  2  r), 

and  the  pressure  lost  in  transmission  will  be  2  v  x  I. 

These  relations  are  simple,  and  such  as  would  be 
obtained  if  continuous  current  was  being  transmitted 
instead  of  alternating  ;  but  if  the  conductors  are  of  com- 
paratively large  diameter,  there  will  be  an  inductive 
effect  which,  by  causing  an  unequal  distribution  of  current 
in  the  conductors,  leads  to  the  apparent  resistance,  and  the 
loss  of  power,  being  greater  with  a  rapidly  alternating 
current  than  with  one  that  is  steady  and  unidirectional. 
This  is  known  as  the  "  skin  effect,"  and  it  becomes  of 
some  practical  importance  when  the  conductors  are  of 
large  diameter,  especially  if  the  frequency  is  high.  As  a 
rule  the  skin  effect  can  be  neglected ;  but  in  refined  calcu- 
lations the  necessary  correction  should  be  made. 

When  a  wire  carries  an  alternating  current,  this  current 
produces  a  varying  flux  of  induction,  not  only  in  the 
medium  surrounding  the  wire,  but  also  in  the  space 
occupied  by  the  material  of  the  conductor  itself.  This 
alternating  flux  has  the  effect  of  crowding  the 
current  toward  the  surface  of  the  conductor,  with  the 
result  that  the  I  R  drop  of  pressure  is  greater  than  it 
would  be  with  a  uniform  current  density.  In  fact,  the 
result  is  the  same  as  if  a  uniformly-distributed  current 


TRANSMISSION   BY   SINGLE-PHASE  CURRENTS      147 


85 


80 


75 


70 


65 


60 


55 


50  Ut  — 


45 
40 
35 
30 
25 
20 
15 
10 
5 
0 


/ 

H 

X 

CO 

I 

. 

/ 

• 

/ 

' 

/ 

1 

/ 

/ 

1  1  1 

"  conductor  in  sauare 

/ 

7 

/ 

/ 

/ 

aofcoppei 

•2 

Z 

1 

/' 

/ 

[ 

Ccrrfi 

'nurn 

'.f  r// 

rren 

t    ; 

Afternat/'riff  current 

1-00  1-02  1'04  1-06  1-08  MO  1-12  1'14  M6  1'18  1'2  1'22  1'24 
FIG.  58. 


1-28  1-3 


were    flowing    in    a    conductor    of    which    the    resist- 
ance  is    R'  ohms  instead  of  R   ohms;    and   the   ratio 

apparent  resistance       R'  .    .. 

—. r-       —,  or^  is  the  "  skin  effect  '  coefficient 

actual  resistance          R 


148  POWER   TRANSMISSION 

of  which  the  value,  for  copper  conductors,  may  be  obtained 
from  the  curve  Fig.  58.  This  coefficient  is  a  function  of 
the  product  section  of  (circular}  conductor  x  frequency,  and 
this  is  the  quantity  which  has  been  used  for  plotting  the 
vertical  ordinates  of  the  curve. 

If  the  conductor  is  not  of  copper,  but  of  any  other 
"  non-magnetic  "  material,  of  circular  cross-section,  the 
curve  Fig.  58  can  be  used  if  the  figures  on  the  vertical 
axis  are  understood  to  be  the  quantity 

conductivity  of  metal  used 
area  x  frequency  >     —  ^^  «,/  copper 


58.  Effect  of  Inductive  Load  on  Line  Losses.  — 

Let  us  still  assume  the  line  to  be  without  inductance  or 
capacity,  and  see  what  is  the  effect  on  the  line  losses 
if  the  "  load  "  at  the  distant  end  has  a  power  factor  less 
than  unity,  such  as  would  be  the  case  if  the  current 
is  supplied  to  induction  motors.  Assuming  a  lag  of 
37  time-degrees,  which  corresponds  to  a  power  factor 
(cos  37  degrees)  of  about  -8,  the  total  current,  to  deliver 
the  same  power  at  the  same  pressure,  will  be  1*25  times 
greater  than  if  the  power  factor  were  unity  ;  and  since, 
for  the  same  loss  of  power  in  the  line,  the  resistance 
multiplied  by  the  square  of  the  current  must  remain  con- 
stant, it  follows  that  the  cross-section  of  the  conductors 
must  vary  inversely  as  the  square  of  the  power  factor— 
the  proper  correction  being  made  for  skin  effect  if  of 
sufficient  importance.  In  the  case  under  consideration, 
the  cross-section  would  have  to  be  increased  in  the  ratio 
of  i  to  '64,  which  corresponds  to  an  increase  in  weight  of 
copper  of  over  50  per  cent.  It  does  not  follow  that  it 
would  be  economical  to  increase  the  section  to  so  great 
an  extent,  but  the  importance  of  a  high  power  factor 
must  not  be  overlooked.  • 


SELF-INDUCTION   OF   THE   LINES 


149 


59.  Effect  of  Taking  into  Account  the  Self- 
induction  of  the  Lines. — Since  there  must  be  a  certain 
distance  separating  the  outward-going  and  return  wires 
of  the  alternating-current  transmission  line,  there  must 
of  necessity  be  an  E.M.F.  of  self-induction  produced  by 
the  alternating  magnetic  flux  in  the  space  between  the 


---"^ "" 


FIG.  59. 


wire ;    and    this   will    be    directly   proportional    to   the 
amount  of  current  in  the  line. 

In  Fig.  59,  O  E'  is  the  pressure  at  receiving  end,  and 
O  I  the  current,  lagging  behind  the  voltage  vector  by  an 
angle  B.  The  vector  O  e,  drawn  in  phase  with  the  current, 
and  equal  in  magnitude  to  2  r  x  I,  is  the  E.M.F.  com- 
ponent required  to  overcome  the  resistance  of  the  line. 
The  counter  E.M.F.  of  self-induction  will  lag  behind  the 


150  POWER   TRANSMISSION 

current  by  a  quarter  of  a  period,  and  the  vector,  O  L, 
has  been  drawn  90  degrees  in  advance  of  the  current  to 
represent  the  component  of  the  total  pressure  at  gener- 
ating end  necessary  to  balance  the  E.M.F.  of  self-induc- 
tion. The  total  requisite  initial  pressure  is  evidently 
O  E,  obtained  by  compounding  the  forces  O  E',  O  e,  and 
O  L.  The  dotted  lines  show  the  effect  of  resistance 
drop  combined  with  inductive  drop  for  the  same  current 
in  the  line,  and  the  same  total  amount  of  power  trans- 
mitted, when  the  power  factor  at  receiving  end  is  unity — 
i.e.,  when  0  =  o.*  A  study  of  this  diagram  (Fig.  59) 
leads  to  the  following  conclusions : 

(1)  The  additional  loss  of  pressure  due  to  the  self-induc- 
tion of  the  lines  is  of   considerably  greater  importance 
on  an  inductive  than  on  a  non-inductive  load. 

(2)  For  a  given  current  and  I2  v  loss  in  the  line,  the 
difference  in  the  power  factors  at  the  two  ends  is  greatest 
when  the  load  is  non-inductive. 

60.  Predetermination  of  Inductive  Drop.— In  the 

diagram  Fig.  59  we  have  assumed  the  inductive  pressure 
drop — represented  by  the  vector  O  L— to  be  known. 
In  order  to  calculate  it,  we  must  know  the  coefficient 
of  self-induction  of  the  circuit  formed  by  the  go  and 
return  wires.  This  will  depend  not  only  upon  the  dis- 
tance separating  the  wires,  but  also  upon  their  diameter. 
The  greater  the  distance  between  wires,  and  the  smaller 
their  diameter,  the  larger  will  be  the  amount  of  magnetic 
flux  produced  by  the  passage  of  unit  current. 

The  coefficient  of  self-induction,  L,  for  one  conductor 

*  This  method  of  drawing  the  diagrams,  which  assumes  the 
current  constant  and  the  pressure  variable,  to  comply  with  the  con- 
dition of  constant  power  at  different  power  factors  is  sometimes 
convenient,  but  the  construction  can  be  modified  to  suit  the  con- 
dition of  a  definite  initial  pressure. 


PREDETERMINATION   OF   INDUCTIVE   DROP      151 

of  an  overhead  transmission  line  ojie  mile  long  is  given 
by  the  formula 

Millihenrys  =  741  x  Iog10    2-568  _ 


where  D  is  the  distance  between  the  centres  of  the  outward 
and  return  conductors,  and  d  is  the  diameter  of  one  con- 
ductor ;  these  measurements  being  expressed  in  the  same 
units. 

Now,  the  counter  E.M.F.  of  self-induction,  in  volts,  on 
the  sine  wave  assumption  is 

EL  =  2  TT/L  x  I, 

where  L  is  the  inductance  in  henryst  and  I  is  the  current 
in  amperes. 

Inserting  the  above  value  of  L,  we  get, 

Volts  induced  per  mile  1  =         6  fi  f  j  l     (2    68^\ 
of  single  conductor  }  -\7/' 

which  enables  us  to  calculate  the  length  of  the  vector 
O  L  in  Fig.  59.  It  must  be  understood  that,  since  this 
formula  takes  into  account  the  reactance  of  one  conductor 
only,  the  value  of  the  voltage  so  obtained  must  be 
doubled  in  order  to  arrive  at  the  total  counter  E.M.F. 
of  self-induction  in  the  loop  of  a  single-phase  transmission 
one  mile  long.  For  any  other  length  of  line  it  is,  of 
course,  merely  necessary  to  multiply  by  the  distance  of 
transmission  expressed  in  miles. 

61.  Capacity  of  Transmission  Lines.  —  In  the  case 
of  a  concentric  or  other  underground  cable  conveying 
alternating  currents,  the  capacity  effects  may  be  very 
great.  A  large  capacity  is  generally  objectionable  in 
practical  working,  although  the  effects  of  capacity  tend 


<  > 

<  > 


154  POWER   TRANSMISSION 

to  balance  those  of  self-induction,  and  so  bring  the  power 
factor  nearer  to  unity.  If  C  is  the  capacity  in  micro- 
farads of  any  system  of  two  conductors,  /  the  frequency, 
and  E  the  potential  difference  between  the  conductors, 
then  the  charging  current — i.e.,  the  alternating  current 
which  will  pass  between  them  irrespective  of  the  load 
at  the  distant  end — will  be  (see  article  26,  Chapter  II., 
p.  60) 

Ic=27T/CE-r    1,000,000, 

on  the  assumption  that  the  wave  form  of  the  E.M.F.  is  a  sine 
curve.  This  current  is  in  quadrature  with  the  E.M.F., 
being  go  degrees  in  advance  of  the  applied  potential  differ- 
ence, or  90  degrees  behind  the  condenser  E.M.F. 

The  above  formula  enables  us  to  calculate  the  capacity 
current  on  the  assumption  of  the  E.M.F.  following  the 
simple  harmonic  law  of  variation,  provided  the  capacity, 
C,  can  be  correctly  predetermined,  or  ascertained  by 
actual  measurement ;  but  when  we  come  to  consider  the 
various  wave  forms  found  in  actual  practice,  it  is  not 
possible  to  calculate  the  capacity  current  with  a  high  degree 
of  accuracy.  The  sine  curve  wave  will  be  found  to  give 
the  smallest  condenser  current  of  any  possible  wave  form, 
while  irregular  wave  forms  may  give  rise  to  capacity 
effects  which  it  is  rarely  possible  to  predetermine  with 
accuracy. 

The  two  diagrams  Figs.  60  and  61,  reproduced  by 
kind  permission  of  Mr.  A.  Whalley,  of  the  British  Insu- 
lated and  Helsby  Cables,  Limited,  show  the  extraordinary 
distortion  of  the  wave  forms  due  to  the  introduction  of 
capacity  in  the  circuit  of  an  alternator  which  does  not, 
under  all  conditions  of  load,  give  a  difference  of  potential 
at  the  terminals  following  the  simple  harmonic  law  of 
variation. 


CAPACITY  OF   TRANSMISSION    LINES  155 

In  Fig.  60,  V  is  the  pressure  and  A  the  current  when 
the  alternator  is  supplying  29-5  amperes  at  2,040  volts  to 
a  transformer  on  a  nearly  non-inductive  load  ;  whereas 
Fig.  6  1  shows  the  altered  pressure  curve  and  the  result- 
ing charging  current  when  the  same  alternator  is  supply- 
ing a  current  of  6-04  amperes  at  1,980  volts  between  the 
two  conductors  of  an  insulated  cable  having  a  capacity 
of  4'9  microfarads.  It  is  interesting  to  observe  how 
the  smallest  ripple  in  the  original  curve  may  become 
distorted  and  magnified.  The  fact  that  the  maximum 
values  of  the  E.M.F.  (and,  therefore,  of  the  charge) 
invariably  occur  at  the  instant  when  the  current  is 
changing  its  direction  should  also  be  noted.  (See 
Fig.  23,  Chapter  II.,  article  26). 

In  the  case  of  overhead  lines,  the  capacity  effects  are 
generally  small,  and  sometimes  negligible.  A  convenient 
formula  for  use  in  calculations  on  overhead  transmission 
lines  is 

C  =   'OI94 


where   C  =  capacity   in    microfarads   between   the   two 
parallel    wires    of    a    single-phase    trans- 
mission one  mile  long  ; 
D  =  distance  between  centres  of  conductors,  in 

inches  ; 

v  =  radius   of   cross-section  of   the  (cylindrical) 
wires  in  inches. 

This  formula  is  not  scientifically  correct,  and  is  not 
applicable  to  parallel  wires  of  which  the  separation,  D, 
is  small  in  proportion  to  the  radius,  v  ;  but  with  the 
usual  separation  between  conductors  on  practical  over- 


156 


POWER  TRANSMISSION 


head   transmissions   no  greater  approach  to  theoretical 
accuracy  is  necessary. 

62.  Vector  Diagram  for  Single-Phase  Trans- 
mission Line,  taking  into  Account  Resistance, 
Inductance,  and  Capacity.— In  Fig.  62,  the  generator 
is  shown  supplying  the  total  current,  I,  at  the  initial 
pressure,  E,  through  the  transmission  lines  to  the  distant 
end,  where  the  load  is  supposed  to  be  partly  inductive. 
The  current  delivered  at  the  end  of  the  line  is  I',  and  the 
pressure  E' ;  the  coefficient  of  self-induction  of  the  trans- 
mission lines  is  2  L,  and  their  resistance  2  r.  It  will 


FIG.  62. 

be  noticed  that  the  whole  of  the  capacity  is  shown  as 
being  concentrated  at  the  end  of  the  line.  This  assump- 
tion leads  to  a  calculated  capacity  current  somewhat  in 
excess  of  what  would  be  obtained  with  the  same  total 
capacity  distributed  along  the  line,  but  it  simplifies  the 
vector  diagram.  A  closer  approximation  to  actual  con- 
ditions would  be  obtained  by  supposing  the  capacity  to 
be  concentrated  at  the  centre  of  the  line  (or,  perhaps, 
rather  nearer  the  receiving  than  the  transmitting  end) ; 
or  if  still  greater  accuracy  be  required,  no  difficulty  need 
be  experienced  in  drawing  the  diagram  on  the  supposition 
that  the  capacity  is  due  to  two  or  more  smaller  condensers 


SINGLE-PHASE  TRANSMISSION  LINE  !$/ 

connected  between  the  lines  at  different  points.  But  it 
must  be  remembered  that  it  is  not  possible  to  calculate 
the  capacity  current  on  a  long  transmission  line  with 
great  accuracy,  because  it  will  depend  largely  upon 
the  E.M.F.  wave  form,  which,  even  if  known  under 
certain  specified  conditions,  will  generally  vary  consider- 
ably with  alterations  in  the  load. 


FIG.  63. 

Assuming,  then,  for  the  sake  of  greater  simplicity  of 
construction,  that  the  total  capacity,  C,  of  the  line  is  con- 
centrated at  the  distant  end,  we  may  proceed  to  draw  the 
vector  diagram  (Fig.  63).  Here  O  E'  and  O  I'  repre- 
sent, respectively,  the  potential  difference  and  the  current 
at  the  distant  end  of  the  line,  the  two  vectors  being  drawn 
with  an  angle  6  between  them,  such  that  cos  0  is  equal 
to  the  assumed  power  factor  of  the  load.  Referring  to 
Fig.  62,  it  will  be  seen  that  a  condenser,  of  capacity  C, 


158  POWER  TRANSMISSION 

is  supposed  to  connect  the  wires  at  the  distant  end ;  the 
impressed  volts  at  condenser  terminals  are  therefore  E', 
and  the  vector  for  the  condenser  current  (equal  to 
2  TT/C  E',  on  the  assumption  of  the  sine  curve  wave  form) 
must  be  drawn  go  degrees  in  advance  of  O  E'.  It  is 
represented  by  O  lc  in  Fig.  63.  The  total  currrent  sup- 
plied to  the  line  at  the  generator  end  will  be  O  I,  obtained 
by  compounding  O  V  and  O  lc.  With  regard  to  the 
E.M.F.  at  generator  end,  this  is  the  resultant  of  three 
components — namely :  O  E',  the  pressure  available  at 
receiving  end ;  E'  a  (drawn  parallel  to  O  I  and  equal 
to  I  x  2  r),  to  compensate  for  the  drop  due  to  resistance ; 
and  a  E  (drawn  at  right  angles  to  O  I  and  equal  to 
27T/x  2  L  x  I),  to  represent  the  pressure  required  to 
balance  the  E.M.F.  of  self-induction.  This  gives  us  O  E 
as  the  necessary  pressure  at  generating  end. 

The  conclusions  to  be  drawn  from  this  diagram  (which 
takes  both  self-induction  and  capacity  into  account)  are 
as  follows : 

(1)  On  an  inductive  load  the  current,  I,  put  into  the 
line  at  the  generating  end  may  be  less  than  the  current,  I', 
at  receiving  end. 

(2)  The  total  current,  I  (under  the  condition  of  a  partly 
inductive  load),  comes  more  and  more  nearly  in  phase 
with  the  E.M.F.  as  the  capacity  current  increases  aip  to 
a  certain  limit  depending  upon  the  power  factor  of  the 
load.     Thus,  if  the  capacity  current  were  equal  to  O  S 
instead  of  O  lci  the  total  current — as  indicated  by  the  chain 
dotted   lines   in   Fig.  63 — would   be  in  phase  with  E'. 
This  illustrates  the  effect  of  capacity  supplying  the  mag- 
netising current  of  the  inductive  apparatus  constituting 
the  load,  and  so  improving  the  power  factor. 

The  charging  current  on  overhead  lines  is  generally 
small.  Let  us  take  as  an  example  the  data  which  follow  : 


CAPACITY  CURRENT  IN   TRANSMISSION   LINE      159 

The  pressure  E'  =  20,000  volts. 
The  frequency  /=  25. 
The  length  of  the  line  =  50  miles. 
The  diameter  of  conductors  =  -36  in. 
Their  distance  apart  =  3  ft. 

Using  the  formula  given  above  for  the  capacity  of  an 
overhead  line,  we  have 

•0194 
Capacity  in  microfarads  per  mile  = }      -5/^g 

=  -00843, 
which  gives  us  for  the  total  capacity 

C  =  50  x  '00843 
=  '4215  microfarads 

The  capacity  current  will  be 

lc  =  2  TT  x  25  x  '4215  x  20,000  x  10-6 
=  1*3  amperes, 

which  is  a  very  small  percentage  of  the  total  current 
(assumed  to  be  about  50  amperes).  At  the  same  time, 
we  have  only  to  suppose  an  increase  in  the  length  of  line 
from  50  to  200  miles,  with  a  correspondingly  higher 
pressure  of,  say,  80,000  volts,  and  imagine  the  frequency 
to  be  50  instead  of  25,  in  order  to  obtain  a  calculated 
capacity  current  of  nearly  42  amperes,  which  is  by  no 
means  negligible.  As  a  matter  of  fact,  the  capacity 
would  be  somewhat  smaller  than  the  value  previously 
calculated,  because  of  the  necessary  increased  spacing 
between  wires. 

It  will  be  understood  that,  where  insulated  cables 
are  used  (only  possible  for  low  pressures  and  short 
distances),  the  capacity  current  may  be  from  twenty  to 


160  POWER   TRANSMISSION 

thirty  times  as  great  as  in  the  case  of  an  equivalent  over- 
head transmission,  and,  moreover,  the  dielectric  losses 
may  become  important. 

63.  Rise  of  Pressure  at  the  Distant  End  of  a 
Long:  Transmission  Line. — A  slight  modification  of 
the  diagram  Fig.  63  will  clearly  show  the  well-known 
effect  of  a  rise  of  pressure  occurring  at  the  far  end  of  a 
line  when  the  load  is  very  small,  provided  also  that  both  self- 
induction  and  capacity  are  present.  We  shall  still  suppose 
the  arrangement  to  be  as  shown  in  Fig.  62,  with  the  one 
exception  that  the  transformer  and  load  at  the  receiving 
end  are  entirely  disconnected.  We  shall  also  suppose  the 

A 

R 


E' 


FIG.  64. 


capacity  and  self-induction  to  be   greater  than   in   the 
previous  example,  so  as  to  magnify  the  effect. 

Draw  O  E'  in  Fig.  64  to  represent  the  pressure  at  the 
distant  end ;  then  O  lc  (at  right  angles  to  O  E' — in  the 
forward  direction — and  equal  to  2  TT/C  E')  is  the  capacity 
current  which,  under  the  condition  of  open  circuit  at 
distant  end,  is  also  the  total  current  flowing  in  the  wires. 
We  now  proceed  with  the  construction  exactly  as  in 
Fig.  63,  making  E'  a  (in  phase  with  the  current)  equal  to 
Ic  x  2  r,  and  a  E  (at  right  angles  to  the  current)  equal 
to  2  TT  /  x  lc  x  2  L.  The  vector  O  E,  which,  it  will  be 
seen,  is  smaller  than  O  E7,  is  the  necessary  pressure  at 
generating  end.  It  will  be  noted  that,  if  self-induction 


RISE  OF   PRESSURE  AT   END  OF   LINE          l6l 

were  absent,  the  pressure  at  generating  end  would  be 
O  a,  which  is  practically  equal  in  length  to  O  E' ;  but, 
since  the  E.M.F.  of  self-induction,  O  L  (for  this 
particular  state  of  things),  is  in  phase  with  E',  it  stands  to 
reason  that  the  necessary  pressure  at  generating  end  will 
be  reduced  accordingly. 

Rises  of  pressure  at  the  end  of  long  underground 
feeders  are  by  no  means  uncommon,  and  they  may  also 
occur  on  long-distance  overhead  lines.  Indeed,  a  rise  'of 
8  per  cent,  has  been  observed  on  a  4o-mile  25,ooo-volt 
line  when  the  circuit  was  open  at  the  distant  end.  It 
should  be  mentioned  that  the  pressure  rises  due  to  the 
combined  effects  of  capacity  and  self-induction  are  often 
greater  in  practice  than  would  be  indicated  by  the 
construction  of  a  diagram  such  as  Fig.  64.  But  this 
diagram  is  based  on  the  assumption  of  the  E.M.F.  wave 
being  a  sine  curve,  and  the  abnormal  effects  sometimes 
met  with  in  practice  are  often  due  to  peaked  or  irregular 
wave  forms.  These  may  lead  to  enormously  increased 
capacity  effects,  which,  unfortunately,  cannot  be  pre- 
determined accurately. 

The  calculation  of  abnormal  voltages  due  to  the 
interruption  of  heavy  currents,  or  to  the  condition  of 
resonance,  would  be  out  of  place  in  this  book,  and  the 
reader  who  is  interested  in  these  matters  is  referred  to 
more  advanced  books,  such  as  "  Transient  Phenomena," 
by  Dr.  C.  P.  Steinmetz,  in  which  these  problems  are 
adequately  dealt  with.  All  that  the  present  writer  can 
attempt  in  these  pages  is  to  describe  in  a  few  words  the 
conditions  that  lead  to  abnormal  rises  of  pressure.  In  the 
first  place  it  should  be  noted  that  energy  can  be  dissipated 
in  a  transmission  line  only  in  the  form  of  heat,  either  in 
the  wire  itself  through  ohmic  resistance,  or  in  the 
surrounding  air  through  corona  loss,  not  to  mention  the 

ii 


162  POWER   TRANSMISSION 

small  leakage  loss  that  occurs  over  insulators.  Now, 
when  a  circuit  has  inductance  or  capacity,  or  both, 
energy  is  stored  in  the  circuit  so  long  as  a  current  is 
flowing  ;  and  this  energy  must  be  given  up  or  dissipated 
when  the  current  ceases  to  flow.  In  the  case  of  an 
alternating  current  in  a  circuit  of  negligible  resistance, 
when  the  current  passes  through  zero  value  all  the 
energy  is  stored  in  the  electrostatic  field,  and  at  the 
instant  when  the  pressure  wave  passes  through  zero 
value  this  energy  is  necessarily  in  the  electro-magnetic 
field.  Thus  there  is  an  oscillation  of  electric  energy,  the 
frequency  of  which  can  be  shown  to  be 

f=      _1  _ 

~  ' 


where  L  and  C  are  expressed  in  henrys  and  farads 
respectively.  This  quantity  is  known  as  the  natural 
frequency  of  the  circuit.  It  will  probably  be  admitted 
without  proof  that,  if  we  impress  on  the  circuit  an  E.M.F. 
of  which  the  periodicity  is  the  same  as  this  "natural 
frequency,"  abnormal  results  may  be  expected.  This  is 
the  condition  of  resonance.  In  practice  the  normal 
frequency  of  transmission  is  much  lower  than  the  natural 
frequency  of  the  line  ;  but  the  presence  of  ripples  in  the 
wave  form  —  which,  by  analysis,  can  be  considered  as 
"higher  harmonics"  —  may  lead  to  the  condition  of 
resonance  and  abnormal  voltages. 

The  oscillations  of  energy,  previously  referred  to,  in  a 
circuit  of  negligible  resistance,  leads  to  the  simple 
relation 


This  quantity   has  been    named   by   Dr.   Steinmetz  the 
"natural  impedance"  of  the  circuit;  and  it  provides  a 


TRANSMISSION   BY   TWO-PHASE   CURRENTS      163 

ready  means  of  calculating  the  maximum  value  of  any 
surge  pressure  due  to  the  sudden  interruption  of  current 
through  the  opening  of  the  circuit.  It  can  be  shown 
that,  on  practical  overhead  lines,  the  surge  pressure,  in 
volts,  cannot  exceed  about  200  times  the  amount  of  the 
current,  in  amperes. 

64.  Transmission  by  Two-Phase  Currents. — Let 

us  assume,  in  the  first  place,  as  when  considering  single- 
phase  transmission,  that  not  only  the  load  at  the  distant 
end,  but  also  the  transmission  line,  are  without  self- 
induction  or  capacity. 

If  we  run  four  conductors  as  indicated  in  Fig.  65, 


FIG.  65. 

keeping  the  two  phases  entirely  separate,  the  arrange- 
ment resolves  itself  into  the  transmission  of  two  distinct 
single-phase  currents ;  and  if  the  load  is  equal  on  both 
sections,  the  total  power  transmitted  will  be 

W  =  2  E  x  I 
or,  2  x   I2  x  (R  +  2  r), 

where  E  =  the  terminal  pressure  of  each  section  and 
I  =  the  current  in  each  section. 

The  pressure  lost  in  transmission  will  be  2  r  x  I, 
where  r  is  the  resistance  of  a  single  conductor ;  and  the 
necessary  weight  of  copper  in  the  conductor,  for  a  given 
pressure  drop  (or  loss  of  power),  will  be  the  same  as  for 


164  POWER   TRANSMISSION 

a  single-phase  transmission  scheme.  It  is,  however, 
evident  that,  by  combining  two  of  the  conductors  to  form 
a  common  return,  the  transmission  of  two-phase  currents 
can  be  effected  with  only  three  wires,  as  shown  in  Fig.  66. 
The  vector  diagram  for  such  a  system  of  transmission 
is  very  simple,  if  we  can  assume  the  resistance,  /,  of  the 
common  conductor  to  be  negligible,  and  it  has  been  drawn 
in  Fig.  67.  Here  the  phase  difference  of  a  quarter 
period  between  the  two  E.M.F.s  is  represented  by  Ex 
being  drawn  90  degrees  in  advance  of  E2,  and  since  the 


FIG.  66. 

circuits  are  balanced — i.e.,  since  the  load  is  the  same  on 
both  phases — we  may  write 


and  as  we  are  not  taking  into  account  either  capacity  or 
self-induction,  the  two  currents  will  be  in  phase  with  the 
respective  E.M.F.s,  and  the  return  current  in  the  common 
conductor  will  be  \/2  times  Ij  or  12. 

When  the  common  return  wire  has  appreciable  resist- 
ance, as  would  be  the  case  on  a  power  transmission 
scheme  by  two-phase  currents,  complications  arise  which 
are  accentuated  by  the  unsymmetrical  phases  of  the 
counter  E.M.F.s  of  self-induction.  The  result  is  that, 


TRANSMISSION   BY   TWO-PHASE   CURRENTS      165 


even  when  the  impressed  E.M.F.s  are  exactly  equal, 
with  a  phase  difference  of  a  quarter  period,  the  currents 
in  the  two  phases  may  not  be  equal,  and  the  phase 
angle  between  them  is  usually  greater  than  90  degrees. 
These  reasons  account  for  the  fact  that  two  -  phase 
currents  are  almost  invariably  transmitted  over  four 
wires.  Taking  into  account  the  additional  fact  that 


FIG.  67. 

transmission  by  three-phase  currents  has  now  been 
adopted  almost  to  the  exclusion  of  two-phase  currents, 
it  appears  unnecessary  to  investigate  further  the  reasons 
leading  to  this  loss  of  symmetry  under  load  conditions, 
and  we  will  pass  on  to  the  consideration  of  power  trans- 
mission by  three-phase  currents. 

65.  Transmission   by  Three-Phase  Alternating 
Currents. — We  shall  again  assume,  in  the  first  instance, 


166  POWER   TRANSMISSION 

that  the  line  is  without  self-induction  or  capacity.  If  we 
were  to  run  six  separate  conductors — i.e.,  two  for  each 
phase — we  should  merely  be  transmitting  three  indepen- 
dent single-phase  currents,  the  arrangement  being  as 
shown  in  Fig.  68 ;  and  if 

e  =  the  potential  difference  at  the  terminals  of  each 

circuit ; 

I  =  the  current  in  each  wire  ; 
v  =  the  resistance  of  each  wire  ; 
R  =  the  resistance  of  the  "  load  "  on  each  phase, 

then  the  total  power  transmitted  would  be 

W  =  3  (e  x  I), 
or,  W  =  3  x   I2  x  (R  +  2  r). 

The  pressure  lost  in  transmission  would  be  2  r  x  I,  and 
the  total  power  lost  in  the  lines  would  be  3  x  I2  x  2  v. 


FIG.  68. 

Let  us  now  see  what  is  the  effect  of  providing  a 
common  return  for  these  three  circuits.  The  arrange- 
ment is  shown  in  Fig.  69. 

The  pressure  at  generating  end  between  the  three 
terminals  of  the  alternator  and  the  common  return,  or 
neutral  point,  is  still  e  volts,  and  the  total  power  trans- 
mitted is  still  W  =  3  (e  x  I) ;  but,  owing  to  the  fact 
that  the  sum  of  the  three  outgoing  currents  is  zero  (seeing 


TRANSMISSION   BY   THREE-PHASE  CURRENTS      167 

that  they  differ  in  phase  by  120  degrees,  as  shown  in 
Fig.  70,  and  that  any  one  current,  such  as  O  B,  is  exactly 
equal  and  opposite  to  the  resultant  of  the  other  two 
currents),  there  will  be  no  current  flowing  in  the  common 
return  conductor,  and  it  follows  that  both  pressure  drop 
and  I2  r  losses  in  the  lines  are  reduced  to  one- half  of.  what 
they  were  with  the  arrangement  of  three  separate  circuits, 
the  power  loss  in  the  lines  being  now  3  I2  r.  This 
clearly  shows  how  the  transmission  by  three-phase 
currents  is  more  economical  as  regards  line  losses  than 
single-phase  transmission.  But  it  must  not  be  over- 


FIG.  69. 

looked  that,  in  order  to  obtain  a  reduction  by  half  of  the 
weight  of  copper  in  the  lines,  the  pressure  between  wires 
is  greater  on  the  three-phase  system  than  on  a  single- 
phase  system  transmitting  the  same  power.  Thus,  the 
pressure,  E  (Fig.  69),  between  any  two  of  the  three 
transmission  wires  is  equal  to  e  x  \/3,  as  shown  by  the 
diagram  Fig.  71.  Here  the  E.M.F.s  in  the  three  sections 
of  the  alternator  windings  are  represented  by  O  A,  O  B, 
and  O  C;  and  since  the  E.M.F.,  E,  between  any  two 
terminals,  such  as  B  and  C  (Fig.  69),  is  the  resultant  of 
the  E.M.F.s  acting  (away  from  the  common  junction  O) 
in  the  two  windings  O  B  and  O  C  connected  in  series, 


168  POWER  TRANSMISSION 

we  must  subtract  one  of  these  (the  vector  O  C)  from  the 
other  (the  vector  O  B).  Thus  the  resultant,  O  E 
(Fig.  71),  is  obtained  by  adding  to  the  vector  O  B  an 
imaginary  vector,  O  C',  exactly  equal  but  opposite  to 
O  C.  This  resultant  is  evidently  equal  and  parallel  to 
the  line,  C  B,  joining  the  ends  of  the  two  vectors  O  B 
and  O  C,  and  it  can  easily  be  shown  to  be  exactly 
\/3  times  greater  than  either  of  these  vectors.  We  may, 
therefore,  write 

E  =  1732  e. 


A 


FIG.  70. 


The  power  of  a  three-phase  circuit,  which  is  equal  to 
three  times  e  x   I  ,  can  evidently  also  be  written 


w  =        x 

or,  W  =   V7  E  x  I, 

where  E  is  the  pressure  between  any  two  of  the  three 
wires. 

In  comparing  three-phase  with  single-phase  trans- 
mission on  the  basis  of  the  same  maximum  potential  difference 
between  wires,  we  have,  for  the  total  three-phase  power 

W  =   A/~^  E   x   I, 


TRANSMISSION    BY   THREE-PHASE   CURRENTS      169 
which,  for  equal  power,  single-phase,  would  be  written 
W  =  E  x   (V~3  I). 

Let  r   =  the  resistance  of  each  conductor  on  the  three- 
phase  system,  and 

rl  =  the  resistance  of  each  conductor  on  the  single- 
phase  system, 

then  weight  of  copper,  single  phase  oc   2  x  — 

and  weight  of  copper,  three  phase  oc    3   x  — 

and  the  ratio 

weight  of  copper,  single  phase 
weight  of  copper,  three  phase 

2  r 
may  be  written  - —  . 

But  for  equal  line  losses  we  have 

which  gives  us  r  =  2  rv 

Hence  the  relative  weights  of  copper  are  as  4:3, 
showing  a  saving  of  25  per  cent,  in  favour  of  the  three- 
phase  system. 

66.  Effect  of  Inductive  Load  on  Line  Losses.— 

We  shall  still  neglect  the  capacity  and  inductance  of  the 
lines,  but  briefly  consider  the  effect  of  an  inductive  load 
on  the  vector  diagrams. 

In  Fig.  72,  the  diagram  has  been  drawn  for  the 
condition  of  a  power  factor  of  unity. 

Here  O  A,  O  B,  and  O  C  are  the  three  current  vectors. 


170  POWER   TRANSMISSION 

The  potential  difference  between  wires  is 

ab  =  bc=ca  =  fc. 
The  length  Qa  =  Ob  =  Oc  =  e=    -* 


FIG.  72. 


The  total  power  is  : 
W  = 


x   I 


c 

\ 


=  3  £  x   I 

=  3  (O  A  x   O  a). 

In  Fig.  73,  the  diagram  has  been  drawn  for  an  inductive 
load. 

Here  there  is  a  certain  displacement  of  the  current 
phases  relatively  to  the  E.M.F.  phases.  It  will  be 


EFFECT  OF  INDUCTIVE  LOAD  ON  LINE  LOSSES      I/I 

noticed  that  the  vertices  of  the  E.M.F.  triangle  no  longer 
lie  on  the  current  lines  as  in  the  previous  diagram. 
The  three  current  vectors  still  make  the  same  angle  of 
1 20  degrees  with  each  other,  but  they  have  been  moved 
bodily  round  (in  the  direction  of  retardation)  through  an 


FIG.  73. 

angle  0.  The  total  power  is,  evidently,  no  longer  equal 
to  three  times  O  A  x  O  a,  but  to  3  O  A'  x  O  a,  where 
O  A'  is  the  projection  of  O  A  on  O  a ;  and  cos  6  is  the 
power  factor  of  the  three-phase  circuit. 

It    was    explained    in    connection    with    single-phase 
currents  that,  for  a  given  total  power  to  be  transmitted 


1/2  POWER   TRANSMISSION 

at  a  definite  pressure,  the  weight  of  the  conductors — if 
the  I2  v  losses  are  to  remain  constant — must  vary  inversely 
as  the  square  of  the  power  factor.  This  rule  necessarily 
applies  equally  to  the  case  of  three-phase  transmission. 
Thus,  in  Fig.  73,  the  power  factor  (cos  0)  is  expressed  by 

O  A' 

the  ratio  ;  but,  if  the  power  and  pressure  are  constant, 

O  A 

the  length  O  A'  will  not  alter,  whatever  may  be  the  power 
factor.  Hence  O  A  (the  current  in  any  one  of  the  wires) 
will  vary  inversely  as  the  power  factor,  and — for  a  given 
loss  in  the  line — the  cross-section  of  the  conductor  must 
vary  as  the  square  of  the  current,  i.e.,  inversely,  as  the 
square  of  the  power  factor. 

67.  Self-induction  of  Three-Phase  Lines.— The 
arrangement  of  three-phase  overhead  lines  is  generally 
such  that  the  wires  occupy  the  vertices  of  an  equilateral 
triangle.  Under  such  a  condition,  it  is  evident  that  the 
magnetic  flux  due  to  one  of  the  wires  will  neither  increase 
nor  decrease  the  amount  of  the  induction  through  the  loop 
formed  by  the  other  two  wires.  As  a  matter  of  fact,  if  the 
three  wires  are  arranged  in  any  other  practical  manner, 
the  effect  of  the  induction  due  to  any  one  wire  on  the 
loop  formed  by  the  other  two  wires  is  generally  negligible. 

The  currents  in  the  three  wires  are  equal ;  but  they 
differ  in  phase  by  120  degrees.  If,  therefore,  we  calculate 
— by  means  of  the  formula  in  article  60,  p.  151 — the 
induced  E.M.F.  on  the  assumption  that  each  wire 
produces  its  own  flux  of  magnetism  independently  of 
the  other  wires,  we  have  merely  to  combine  two  such 
E.M.F.s  in  the  manner  previously  indicated  (see  Fig.  71) 
to  obtain  the  total  E.M.F.  in  any  one  of  the  three  loops. 
This  is  evidently  V~$  times  as  great  as  the  induced  E.M.F. 
calculated  for  the  current  in  one  of  the  wires  only. 

By  far  the  simplest  way  of   constructing  the  vector 


SELF-INDUCTION    OF   THREE-PHASE   LINES      1/3 

diagrams  for  three-phase  circuits  is  to  consider  the  three 
phases  independently,  and  Fig.  74  shows  the  effect  of 
taking  into  account  the  self-induction  and  ohmic  resistance 
of  the  lines. 

Here  A,  B,  and  C  are  the  current  vectors,  and  a  b'  cf 


>m 


(e)x 


Xei 


FIG.  74. 

is  the  E.M.F.  triangle  for  the  receiving  end;  the  angle 
of  lag  being  0. 

Consider  O  A  and  O  a'  as  being  the  current  and 
E.M.F.  vectors  of  a  single-phase  circuit.  Draw  a'  in 
parallel  to  O  A  to  represent  the  ohmic  drop  due  to  the 
resistance  of  one  conductor,  and  m  a,  at  right  angles  to 


1/4  POWER   TRANSMISSION 

O  A,  to  represent  the  necessary  impressed  voltage  to 
balance  the  E.M.F.  of  self-induction,  on  the  assumption 
that  the  magnetic  field  is  due  solely  to  the  current,  O  A, 
in  one  conductor.  Join  O  a,  and  note  that,  so  far,  the 
construction  is  exactly  similar  to  that  adopted  in  Fig.  59 
for  a  single-phase  circuit.  We  have  merely  to  suppose 
the  same  construction  to  be  followed  in  the  case  of  the 
remaining  two  phases,  and  the  result  is  the  triangle  a  b  c, 
which  represents  the  necessary  three-phase  applied  E.M.F. 
at  the  generating  end. 

The  conclusions  drawn  from  the  diagram  Fig.  59  when 
treating  of  single-phase  currents  are  equally  applicable 
to  the  case  of  a  three-phase  transmission. 

Assuming  the  distance  between  the  wires  to  be  the 
same  for  a  three-phase  as  for  a  single-phase  transmission, 
let  us  see  whether  either  system  has  an  advantage  over 
the  other  from  the  point  of  view  of  inductive  drop. 

In  the  first  place,  the  total  power  delivered  on  the 
three-phase  system  (see  diagram  Fig.  74)  is 

3  O  A  x  O  a'  x  cos  0 
=  3  I  x  e'  x  cos  B 

I  x  E'  x  cos  B, 


where  I  =  the  current  in  any  one  conductor  and  E'  =  the 
pressure  between  wires  at  receiving  end. 

In  a  single-phase  system  supplying  the  same  power  at 
the  same  pressure  between  wires,  the  current  is,  therefore, 
tj  3  times  as  great. 

The  inductive  drop  in  one  of  the  three-phase  loops  is 
proportional  to  I  x  V  3,  and  exactly  equal  to  the  induc- 
tive drop  due  to  each  of  the  single-phase  wires  ;  hence,  if 
the  diameters  of  the  wires  were  the  same  in  both  cases, 
the  inductive  drop  in  a  three-phase  line  would  be  only 


CAPACITY  IN  THREE-PHASE  LINES         175 

half  that  in  an  equivalent  single-  phase  line.  As  a  matter 
of  fact,  the  diameter  of  the  conductors  in  the  single-phase 
line  would  be  greater,  on  account  of  the  larger  current, 
and  this  has  the  effect  of  reducing  the  self-induction  on  the 
single-phase  line  by  something  under  10  per  cent. 

68.   Capacity    in     Three  -  Phase    Transmission 

Lines.  —  The  capacity  of  a  three-phase  line  with  the 
wires  arranged  in  the  form  of  an  equilateral  triangle  may 
be  considered  as  being  made  up  of  three  equal  capacities, 
all  measured  between  any  one  wire  and  the  line  of  zero 
potential,  which  may  be  taken  as  a  line  passing  through 
the  centre  of  the  triangle.  The  three  capacities  are 
shown  diagrammatically  in  Fig.  75. 

The  capacity  of  each  of  the  three  imaginary  condensers 
shown  in  the  diagram  may  be  calculated  by  the  formula  : 


r  _ 

" 


where  C  =  the  capacity  per  mile  in  microfarads  ; 
D  =  the  distance  between  wires  ; 
r  =.  the  radius  of  any  one  wire. 

It  follows  that  the  capacity  current  per  conductor  can 
readily  be  calculated  exactly  as  explained  in  connection 
with  single-phase  transmission,  the  condenser  E.M.F. 
being  taken  as  the  pressure  between  the  wire  and  neutral 

line,  which  is  equal  to   —  =  times  the  pressure  between 

^3 
wires. 

This  method  of  calculating  the  capacity  effects  on  a 
three-phase  line  is  very  simple  and  convenient  ;  it  is 
evidently  based  on  the  assumption  that  the  influence 


POWER   TRANSMISSION 

of  the  earth  may  be  neglected ;  and  this  assumption  is 
justified  if  the  distance  between  the  wires  is  small  as 
compared  with  their  height  above  ground. 

The  formulae  for  the  capacity  of  three-core  cables  with 
lead  coverings  are  not  so  simple  as  for  overhead  wires, 
and  information  regarding  the  amount  of  capacity  current 
to  be  expected  in  a  given  type  of  cable  should  be  obtained 
from  the  maker. 


Equipotential  Line 


Radius  = 


FIG.  75. 

69.  Most  Economical  Section  of  Transmission 
Lines. — In  the  examples  previously  worked  out,  we 
have  assumed  certain  I2  r  losses  in  transmission  which 
have  enabled  us  to  draw  certain  conclusions  as  regards 
pressure  drop,  phase  displacement,  etc.  We  shall  now 
briefly  consider  the  chief  points  to  be  borne  in  mind 
when  determining  the  most  economical  size  of  conductors 
for  any  given  conditions  of  electric  transmission. 

The  question  we  have  to  solve  is  simply  this  :  given  a 


MOST   ECONOMICAL   SECTION    OF   LINES        177 

certain  system  of  transmission  (whether  single-  or  poly- 
phase), will  it  pay  us  better  to  put  in  large  conductors  at 
a  high  initial  cost,  but  having  small  I2  v  losses,  or  small 
conductors  at  a  lower  initial  cost,  but  having  large  I2  v 
losses  ?  There  is  evidently  a  particular  size  of  conductor 
which  will  prove  to  be  the  most  economical  to  use  on  any 
given  transmission  scheme;  but  the  difficulties  of  correctly 
estimating  the  amount  and  duration  of  load  are  generally 
considerable,  and  we  have  to  be  satisfied  with  a  more  or 
less  close  approximation  to  the  best  results. 

Again,  it  will  sometimes  be  found  that  the  section 
of  conductor,  as  worked  out  in  accordance  with  eco- 
nomical considerations,  cannot  be  adopted  in  practice  for 
either  or  all  of  the  following  reasons : 

(1)  The  current  density  may  be  so  high  as  to  raise  the 
temperature  beyond  the  safe  limit. 

(2)  The  pressure  drop  on  a  long  line  may  be  greater 
than  can  conveniently  be  dealt  with. 

(3)  In  the  case  of  overhead  wires,  the  section  may  be  so 
small  that  the  line  would  be  mechanically  weak. 

(4)  In  the  case  of  extra  high  pressures,  the  surface  of  the 
wire  may  be  so  small  as  to  lead  to  excessive  corona  loss. 

The  corrections  to  be  made  in  such  cases  are  fairly 
obvious  :  the  section  of  the  conductor  must  be  increased 
to  the  required  amount  without  regard  to  the  question  of 
economy  ;  or  it  may  be  found  that  aluminium  conductors 
can  be  adopted  with  advantage.  If  the  cables  are  insu- 
lated and  laid  underground,  and  the  economical  current 
density  is  so  high  as  to  raise  the  temperature  unduly,  two 
or  more  cables  will  have  to  be  laid  side  by  side  and 
connected  in  parallel,  or  the  cross-section  of  the  single 
cable  will  have  to  be  increased,  according  to  which 
method  proves  to  be  the  cheapest. 

Kelvin's  law,  in  its  simplicity  (and,  indeed,  when  the 

12 


178  POWER   TRANSMISSION 

law  of  maximum  economy  departs  from  this  simplicity,  it 
ceases  to  be  Kelvin's  law),  may  be  stated  as  follows  : 
"  The  most  economical  section  of  a  feeder  is  that  which 
makes  the  annual  cost  of  the  I2  r  losses  equal  to  the 
annual  interest  on  the  capital  cost  of  the  copper  in  the 
line,  plus  the  necessary  annual  allowance  for  deprecia- 
tion." The  cross-section  should,  therefore,  be  determined 
solely  by  the  current  which  the  conductor  has  to  carry, 
and  not  by  the  length  of  the  line,  or  an  arbitrary  limit  of 
the  percentage  full  load  drop  in  pressure ;  the  drop,  even 
if  considerable,  must  take  care  of  itself.  If  there  are 
reasons  which  make  a  large  drop  undesirable,  then,  if 
necessary,  economy  must  be  sacrificed,  and  the  line 
calculated  on  the  basis  of  pressure  drop  only. 

Kelvin's  law  is  based  on  the  assumption  that  bare 
conductors  are  used,  and  that  the  cost  of  these,  erected  in 
position,  is  directly  proportional  to  the  weight  of  the 
copper.  This  assumption  is  not  justified  in  practice 
if  insulated  cables  are  used,  the  cost  of  which,  per  pound 
of  copper,  is  greater  for  the  small  than  for  the  large  sizes. 
But,  in  the  case  of  an  overhead  transmission  line  with 
bare  conductors,  Kelvin's  law  may  give  sufficiently 
accurate  results ;  because,  although  the  cost  of  erecting 
the  smaller  wires  may  be  greater  per  pound  than  for  the 
larger  sizes,  the  insulators,  supports,  etc.,  may  become 
more  costly  as  heavier  wires  are  used,  thus  making  that 
portion  of  the  total  cost  of  the  line  which  depends  upon  the 
section  of  the  conductor  approximately  proportional  to  the 
weight  of  the  same. 

To  cover  all  cases  in  practice,  the  law  of  maximum 
economy  may  be  stated  as  follows  :  The  annual  cost  of  the 
energy  wasted  per  mile  of  the  transmission  line,  added  to  the 
annual  allowance  (per  mile)  for  depreciation  and  interest  on  first 
cost,  shall  be  a  minimum. 


MOST   ECONOMICAL   SECTION   OF   LINES 

It  is  not  an  easy  matter  to  estimate,  even  approxi- 
mately, the  probable  amount  of  the  I2  v  losses  during 
the  year's  working.  The  energy  lost  in  a  given  conductor 
depends  upon  the  square  of  the  current  and  the  time 
during  which  the  current  is  flowing.  If  a  cable  conveys 
a  current  I  only  twelve  hours  out  of  the  twenty-four  hours, 
then  the  energy  wasted  per  day  is  only  half  what  it  would 
have  been  had  the  current  been  flowing  continuously,  and 
the  watt-hours  may  be  expressed  as  12  x  I2  x  r,  or  as 

I2 
24  x  —  x  r :  it  follows  that,  in  working  out  the  energy 

2 

lost  per  annum  in  the  conductor,  we  have  to  multiply  the 
resistance,  not  by  the  square  of  the  maximum  current 
which  the  conductor  will  at  times  be  carrying,  but  by  the 
mean  value  of  the  square  of  the  current  throughout  the 
year.  The  expected  average  daily  load  curve  should 
therefore  be  drawn,  and  the  mean  value  of  I2  calculated 
therefrom.  If  the  capacity  current  is  considerable,  this 
must  be  taken  into  account  when  calculating  the  average 
value  of  the  square  of  the  current.  The  diagram  Fig.  76 
will  serve  to  explain  how  the  proper  size  of  conductor 
may  be  calculated. 

Let  us  suppose  that  10  per  cent,  is  to  be  allowed 
for  depreciation  and  interest  on  cost  of  conductor ;  then, 
in  the  diagram  Fig.  76 — where  the  horizontal  distances 
represent  resistance  per  mile,  and  the  vertical  distances 
represent  money — plot  the  curve  A,  which  gives  the  rela- 
tion between  10  per  cent,  of  the  capital  spent  on  the 
conductor,  and  its  resistance  in  ohms  per  mile.  Now 
calculate  the  cost  of  the  I2  v  losses  per  mile  of  cable,  for 
any  particular  value  of  the  resistance,  and  draw  the 
straight  line  O  B  through  the  origin,  which  will  give  us 
the  cost  for  any  other  resistance  of  conductor. 

By  adding  the  ordinates  of  the  curves  A  and  B,  the 


i8o 


POWER   TRANSMISSION 


curve  C  is  obtained,  of  which  the  minimum  value  corre- 
sponds with  the  resistance  per  mile  of  the  conductor 
which  will  be  the  most  economical  to  use,  whatever  may 


Resistance,  Ohms  per  mile,  of  Single  Conductor 
6  10  1-5 


%  'j£ 


X* 


FIG.  76. 


be  the  length  of  the  line  or  the  pressure  required  at  the 
receiving  end. 

Let  us  briefly  consider  how  the  two  costs,  of  which 
the  sum  is  to  be  a  minimum,  should  be  estimated. 


MOST   ECONOMICAL   SECTION    OF   LINES        l8l 

In  the  first  place,  the  capital  outlay  on  which  the 
interest  is  calculated  must  include  the  insulation  on 
cables  (if  any)  and  the  difference  in  cost  (if  any)  of 
the  conduits  or  overhead  construction  which  is  dependent 
upon  the  size  of  the  conductors.  No  capital  charge  which  is 
constant  for  all  sizes  of  conductor  within  practical  limits 
need  be  taken  into  account,  because  the  effect  of  adding 
the  same  length  to  all  the  ordinates  of  the  curve  A 
(Fig.  76)  will  be  to  raise  the  curve  C  by  the  same 
amount,  without  altering  either  its  shape,  or  the  position  (in 
the  horizontal  direction)  of  its  point  of  minimum  value. 

With  regard  to  what  is  a  reasonable  percentage  to 
allow  for  interest  on  cost  of  conductors,  and  depreciation, 
this  will  have  to  depend  on  commercial  considerations 
and  the  estimated  "  life  "  of  the  conductors ;  also  on  the 
probable  scrap  value  of  the  metal  when  it  is  replaced. 

The  second  of  the  two-cost  items — namely,  the  cost  per 
annum  of  the  energy  wasted  in  the  line — is  more  difficult 
to  estimate.  It  will  generally  be  the  works  cost  of  the 
power;  but  it  must  not  be  overlooked  that  there  are 
conditions  under  which  the  cost  of  power  wasted  in  the 
line  is  almost  negligible.  Such  a  condition  occurs  in 
connection  with  hydro-electric  schemes,  when  the  demand 
for  power  is  small  in  relation  to  the  power  available. 
On  the  other  hand,  if  the  demand  exceeds  the  available 
supply,  the  cost  of  the  wasted  power  is  actually  the 
selling  price.  This  was  pointed  out  by  Professor  George 
Forbes  in  his  paper  on  "  Distant  Electric  Power  Trans- 
mission."* 

Example  illustrating  Use  of  Diagram. — Consider  an  over- 
head line  with  bare  copper  conductors,  to  carry  a  maxi- 
mum current  of  60  amperes.  Let  us  assume  that  probable 
load  curves  have  been  drawn,  which  will  enable  us  to 
*  Journ.  Inst.  E.  E.t  vol.  xxix.,  May,  1900. 


1 82  POWER   TRANSMISSION 

estimate  the  Vmean  square  value  of  the  current  through- 
out the  year.  This  might  be  about  one-third  of  the 
maximum  value;  let  us  say  20  amperes.  If,  therefore, 
we  multiply  the  resistance  of  each  conductor  by  the  square 
of  20,  we  shall  obtain  the  total  power  lost  in  the  line  on 
the  assumption  that  this  loss  is  going  on  day  and  night 
throughout  the  year.  This  gives  us  as  the  total  number 
of  Board  of  Trade  units  wasted  in  each  conductor  per 
annum  for  an  assumed  resistance  of  i  ohm  : 

Units  .    I    X    (20)'    X    24    X    365 

I,OOO 

=  3,500 ; 

and  if  the  cost  of  this  power  be  taken  at  id.  per  unit,  the 
cost  would  be 

3>5QQ      .   f      6 

12    X    20         *    ^ 

This  enables  us  to  plot  the  point  p  on  the  curve  O  B, 
and,  by  joining  O  p,  we  can  read  off  the  cost  of  power 
per  annum  for  any  other  resistance  of  conductor. 

With  regard  to  the  curve  A,  we  shall  suppose  that 
10  per  cent,  is  to  be  allowed  for  depreciation  and  interest 
on  capital  spent  on  conductors  ;  and,  if  we  take  the  cost 
of  these  at  is.  3d.  per  pound,  we  can  readily  plot  a 
number  of  points  through  which  this  curve  must  be 
drawn.  The  curve  C,  as  already  explained,  is  the  result 
of  adding  the  ordinates  of  curves  A  and  B.  Its  minimum 
value,  in  our  example,  corresponds  to  a  resistance  of 
•62  ohm  per  mile,  which  gives  us  as  the  most  economical 
conductor  a  stranded  cable  somewhere  between  19/16  and 
19/i5>  or  tne  equivalent  solid  wire.  (It  should  be  noted 
that — since  the  curve  A  is  a  rectangular  hyperbola — the 
minimum  value  of  C  is  on  the  same  ordinate  as  the  point 


ADVANTAGES  OF   DIFFERENT   SYSTEMS         183 

where  the  two  curves  cross — i.e.,  where  A  is  equal  to  B  ; 
but  this  is  not  necessarily  the  case  when  the  conductors 
are  provided  with  an  insulating  covering.) 

It  may  be  well  to  point  out  that  the  graphic  method  of 
applying  Kelvin's  law  is  used  here  mainly  because  it  is 
helpful  in  explaining  the  principles  upon  which  the  eco- 
nomic conductor  section  is  determined.  Other  methods  of 
calculation  would  usually  be  adopted  by  the  practical 
engineer ;  and  these  calculations,  together  with  the 
broader  aspects  of  transmission  line  economy,  are  dis- 
cussed in  the  writer's  book  on  Overhead  Power  Trans- 
mission :  it  would  be  out  of  place  to  go  further  into 
these  matters  here. 

70.  Relative  Advantages  of  Different  Systems 
of  Transmission. — By  way  of  introduction,  it  should 
be  mentioned  that  a  comparison  of  various  systems  on 
general  lines  is  not  only  useless,  but  impossible  :  there  is 
no  basis  of  comparison  applicable  to  all  conditions  of 
practice,  and  each  particular  case  must  be  considered  on 
its  own  merits. 

The  following  points  should  not  be  overlooked  : 

(1)  In  comparing  the  relative  cost  of  conductors  neces- 
sary for  two  different  systems  of  transmission,  the  efficiency 
must  be  the  same  in  both  cases — i.e.,  the  same  amount  of 
energy  must  be  considered  as  being  transmitted  the  same 
distance  with  the  same  loss. 

(2)  What  is  to  be  understood  by  the  pressure  must  be 
clearly  defined.   For  instance,  when  comparing  continuous 
with  alternating  current  transmission,  it  must  not  be  for- 
gotten that  the  stress  on  the  insulation  is  less  with  con- 
tinuous currents  than  with  alternating  currents  at  the 
same  \/mean  square  pressure.     In  fact,  on  the  assump- 
tion of  a  sine  curve  wave,  it  is  easy  to  see  that — for  the 
same  power  transmitted,  at  the  same  efficiency,  and  with 


184  POWER   TRANSMISSION 

the  same  maximum  stress  on  insulation — the  weight  of  copper 
for  a  single  phase  alternating-current  transmission  would 
be  double  that  required  if  continuous  currents  were  used. 

Again,  when  considering  alternating-current  systems, 
it  is  necessary  to  decide  whether  it  is  the  maximum 
potential  above  earth  or  the  maximum  pressure  between 
wires  which  should  be  taken  as  a  basis  of  comparison. 

Assuming  the  middle,  or  neutral  point,  of  the  various 
systems  to  be  "  earthed,"  and  allowing  for  a  definite 
maximum  pressure  between  any  wire  and  earth,  the 
comparison  between  the  different  alternating-current 
systems  becomes  a  simple  matter. 

Let  e  and  I  stand  respectively  for  the  voltage  and 
current  per  leg  in  either  of  the  three  systems  shown 
diagrammatically  in  Fig.  77.  Then  the  total  power 
transmitted  in  each  case  will  be 

e  I  cos  0  x  n, 

where  cos  0,  as  usual,  is  the  power  factor,  and  n  =  the 
number  of  legs. 

If  v  be  the  resistance  of  each  line  conductor,  the  total 
line  loss  will  be 

I2  v  x  n, 

and,  for  the  same  line  efficiency,  the  weight  of  copper  per 
horse-power  transmitted  will  evidently  be  the  same  in  all 
cases. 

This  leads  us  to  the  conclusion  that,  for  any  polyphase 
system,  the  power  lost  in  the  line  depends  only  upon  the 
joint  resistance  of  the  conductors,  the  power  transmitted, 
and  the  power  factor,  provided  the  pressure  between  any  wire 
and  the  neutral  point  is  constant. 

Let  us  neglect  the  capacity  and  self-induction  of  the 


ADVANTAGES   OF   DIFFERENT   SYSTEMS        185 


1 86  POWER  TRANSMISSION 

line,  and  assume  that  cos  9  is  the  power  factor  of  the 
load  at  receiving  end.     Then 

power  delivered  =  n  e  I  cos  0  =  W, 

and  the  power  lost  in  line  =  n  I2  r. 

This  last  quantity,  expressed  as  a  percentage  of  the 

power  delivered,  becomes 

n  I2  r  x  100 

percentage  power  lost  in  transmission  =  -   — ^ • 

Let  us  denote  by  R^  the  joint  resistance  of  all  the  line 
conductors  in  parallel — i.e.,  R^  =  -.     This  gives  us,  for 

the   numerator   of  the   above   percentage   loss,  the  ex- 
pression 

n2  I2  R^  x  100. 

W2 


But  n2  P  is  equal  to  -2 

Hence  we  may  write  : 

Percentage  power  lost   in  trans-  \ 

mission     for     any     balanced  I  =-— ^n>  x  100. 

polyphase  system  J 

It  might  at  first  sight  appear  as  if  there  were  nothing 
to  choose  between  the  various  systems  shown  in  Fig.  77 ; 
but  it  must  not  be  forgotten  that  the  potential  difference 
between  the  wires  has  not  been  taken  into  consideration. 
When  this  is  done,  the  advantage  will  be  clearly  seen  to 
be  in  favour  of  the  three-phase  system ;  thus,  the  maxi- 
mum pressure  between  any  two  of  the  line  wires  will  be 
i '73  e  for  the  three-phase  system,  and  2  e  for  the  single- 
or  four-phase  system. 


ARRANGEMENT  OF   OVERHEAD   LINES          187 

Although  we  have  assumed  the  case  of  an  earthed 
neutral,  there  is  much  to  be  said  in  favour  of  a  system 
without  an  earthed  point,  mainly  as  regards  reduced  risk 
of  stoppage  through  a  breakdown  of  insulation. 

Where  current  for  power  and  lighting  is  to  be  taken  off 
the  same  mains,  the  two-phase  system,  with  the  phases 
kept  entirely  distinct,  appears  to  offer  some  advantages ; 
or  the  power  may  be  transmitted  by  three-phase  currents 
and  converted  into  two-phase  by  a  suitable  arrangement 
of  transformers. 

For  the  transmission  of  energy  to  a  distance  by  means 
of  alternating  currents,  the  three-phase  system  would,  on 
the  whole,  appear  to  have  advantages ;  and  it  has  been 
adopted  on  the  continent  of  America  almost  to  the  ex- 
clusion of  other  systems.  On  the  continent  of  Europe, 
and,  quite  recently,  in  this  country,  power  transmissions 
have  been  carried  out  with  continuous  currents  on  the 
Thury  system,  which,  in  special  cases,  compares  very 
favourably  with  the  more  usual  polyphase  trans- 
missions. 

71.  Arrangement  of  Overhead  Lines  in  Practice. 

— It  is  not  intended  to  deal  with  any  questions  concern- 
ing the  practical  construction  of  transmission  lines,  or  the 
calculation  of  stresses  and  strains  in  wires  or  supports ; 
but  this  chapter  would  not  be  complete  without  some 
mention  of  various  minor  matters  affecting  the  design  of 
transmission  lines  from  an  electrical  standpoint. 

Distance  between  Wires. — The  spacing  of  conductors  on 
overhead  polyphase  transmission  lines  depends  on  the 
voltage  and  the  length  of  span.  A  convenient  rule  for 
spacing,  which  recent  practice  confirms  as  being  reason- 
able and  satisfactory,  is  to  add  1 2  to  the  pressure  between 
wires  expressed  in  kilovolts ;  this  gives  the  distance 
between  wires  in  inches  for  spans  up  to  200  feet.  For 


1 88  POWER   TRANSMISSION 

greater  spans,  up  to  a  limit  of  800  feet,  add  another 
5  inches  for  every  100  feet  length  in  excess  of  200. 

Clearance  to  Pole  or  Tower. — This  should  not  be  less  than 
9  inches  for  pressures  up  to  10,000  volts.  The  clearance 
would  be  increased  to  about  18  inches  at  35,000  volts, 
and  24  inches  at  70,000  volts. 

Clearance  above  Ground. — The  minimum  clearance  be- 
tween overhead  conductors  and  ground  is  usually  about 
20  feet  on  the  lower  voltage  lines,  such  as  would  be 
carried  on  wood  poles ;  and  30  feet  or  more  on  the  extra 
high  pressure  lines  carried  on  steel  towers.  The  actual 
height  of  the  point  of  support  will  depend  on  the  length 
of  span  and  the  diameter  of  the  conductor,  also  upon 
whether  copper  or  aluminium  is  used,  as  these  are  factors 
which  influence  the  "  sag  "  of  the  suspended  wire.  The 
span  will  usually  be  between  the  limits  of  about  150  feet 
on  wood  pole  lines,  and  400  to  600  feet  on  steel  tower 
lines ;  but  considerably  greater  spans  are  used  at  river 
crossings  or  to  meet  special  conditions. 

Aluminium  Conductors. — For  the  same  conductivity  as 
copper,  the  diameter  of  an  aluminium  wire  will  be  about 
one  and  a  quarter  times  as  great.  The  weight  for  the 
same  conductivity  is  almost  exactly  half  that  of  copper, 
but  the  larger  area  exposed  to  high  winds  must  be  taken 
into  account  in  making  strength  calculations.  In  countries 
where  the  price  of  aluminium  is  not  controlled  for  the 
benefit  of  the  copper  market,  the  cost  of  a  transmission 
line  usually  works  out  lower  with  aluminium  than  with 
copper  conductors  ;  but  it  does  not  necessarily  follow 
that  the  cheaper  metal  should  be  adopted.  Hard-drawn 
copper  is  an  excellent  material  for  overhead  lines,  and  it 
is  mechanically  stronger  than  aluminium.  The  relative 
merits  of  the  two  metals  should  be  carefully  considered 
in  connection  with  each  particular  power  scheme.  A 


ARRANGEMENT   OF   OVERHEAD   LINES          189 

very  large  number  of  important  undertakings  have 
adopted  aluminium  in  preference  to  copper. 

Insulators. — These  are  usually  of  the  "  pin  "  type  for 
pressures  below  40,000  volts,  and  of  the  "  suspension  " 
type  for  the  higher  voltages.  In  the  "  suspension  "  type 
a  number  of  separate  unit  insulators  are  strung  in  series 
between  the  point  of  attachment  and  the  conductor, 
which  is  suspended  below  the  cross-arm,  instead  of  being 
above  the  cross-arm  as  with  the  pin  type  insulator. 

Porcelain  is  the  material  most  commonly  used  for  in- 
sulators, but  glass  is  used  occasionally  for  the  lower 
pressures.  The  design  of  insulators  for  the  higher  volt- 
ages cannot  be  undertaken  without  thorough  study ;  and 
modern  development  in  the  manufacture  of  insulators  is 
largely  the  result  of  experience.  Space  does  not  permit 
of  any  discussion  of  the  principles  of  design,  which  are 
the  same  for  insulators  used  on  polyphase  systems  as  for 
those  used  on  any  other  high-voltage  transmission.  It 
may  be  mentioned,  however,  that  the  problems  facing  the 
designer  are  concerned  mainly  with  the  proper  distribu- 
tion of  electrostatic  capacities  between  the  various  parts  of  a 
high-pressure  insulator.  The  leakage  resistance,  whether 
of  surfaces  or  of  the  material  of  the  insulator,  is  a 
secondary  matter. 

Protection  against  Lightning  and  Abnormal  Pressure  Rises. 
— It  is  usual  to  instal  lightning  arresters  at  the  generating 
end  of  the  line,  and  also  at  the  receiving  stations.  On 
long  lines  arresters  are  also  sometimes  distributed  at 
intervals,  as,  for  instance,  at  junctions  or  switching 
points.  It  is  difficult  to  protect  the  line  itself  from  a 
direct  stroke  of  lightning,  but  damage  from  this  cause  is 
rare.  An  earthed  "guard  wire" — which  may  be  a 
stranded  galvanised  steel  cable  about  T5F  inch  in  diameter 
— joining  the  tops  of  the  poles  or  towers,  and  carried  the 


POWER   TRANSMISSION 

whole  length  of  the  line  above  the  conductors,  affords 
good  protection,  and  is  extensively  used,  although  some 
engineers  object  to  its  use  on  the  grounds  of  expense  and 
the  possibility  of  the  wire  falling  across  the  conductors 
and  causing  interruption  to  service.  If  the  guard  wire  is 
not  used,  lightning  rods  should  be  provided  on  poles  or 
towers  at  least  every  500  feet.  The  insulators  them- 
selves may  be  protected  by  "  arcing  rings,"  which  are 
so  placed  that  an  abnormal  rise  of  pressure  will  leap 
across  clear  of  the  porcelain,  and  so  prevent  damage  due 
to  the  heat  of  the  arc.  The  main  object  of  lightning 
protective  devices  is,  however,  to  prevent  damage  to 
apparatus  in  generating-  and  sub-stations. 

The  ordinary  type  of  horn  gap  arrester  may  give  very 
good  results  if  properly  installed,  with  a  non-inductive 
resistance  in  series  with  the  ground  connection ;  but  it 
does  not  cover  the  whole  field  of  lightning  protection, 
and  shares  with  all  spark-gap  devices  the  disadvantage 
that  it  is  liable  to  set  up  surges  or  high-potential  dis- 
turbances in  the  circuit. 

The  multi-gap,  or  so-called  "low  equivalent,"  arrester 
consists  of  many  small  air  gaps  between  cylinders  of 
"  non-arcing  "  metal,  all  in  series.  It  gives  good  results 
on  circuits  up  to  about  30,000  volts. 

For  the  higher  voltages  the  aluminium  cell  arrester  is 
extensively  used,  especially  in  America ;  and  it  certainly 
seems  to  afford  reasonably  good  protection  when  properly 
installed.  A  number  of  aluminium  cups  or  trays,  sepa- 
rated by  an  electrolyte,  are  built  up  in  the  form  of  a 
column  which  forms  a  connection  between  line  and 
ground.  A  film  of  hydroxide  of  aluminium  is  formed 
on  the  plates,  and  this  resists  the  flow  of  current  until  a 
certain  critical  voltage  is  reached,  which  punctures  the 
film  at  a  multitude  of  points,  allowing  a  fairly  large 


ARRANGEMENT   OF  OVERHEAD   LINES          Ipl 

current  to  pass  to  ground ;  but  so  soon  as  the  discharge 
has  taken  place,  and  the  pressure  returns  to  normal,  the 
film  re-forms,  and  is  ready  to  take  another  discharge. 

European  methods  of  lightning  protection — which 
cover  those  installations  carried  out  abroad  to  the  speci- 
fications of  European  engineers — differ  considerably  from 
American  practice ;  but  the  time  is  hardly  ripe  for  passing 
an  unbiassed  judgment  in  favour  of  either  European  or 
American  practice.  The  reason  of  these  differences  of 
method  lies  probably  in  the  fact  that  the  American 
engineer  has  to  take — or  prefers  to  take — what  the  one 
or  two  controlling  manufacturing  concerns  choose  to  offer 
him ;  while  the  European  engineer  is  at  liberty  to  make 
full  use  of  his  knowledge,  experience,  and  judgment.  It 
is  not  suggested  that  the  manufacturing  monopolies  of 
America  are  not  "up-to-date"  in  the  electrical  apparatus 
they  put  on  the  market,  but  merely  that  they  set  the 
fashion  in  the  appliances  used  in  their  country.  It  should, 
indeed,  be  mentioned  that  Professor  E.  E.  F.  Creighton, 
Consulting  Engineer  to  the  General  Electric  Company, 
is  one  of  the  greatest  authorities  on  lightning  protection ; 
and  those  who  wish  to  pursue  this  subject  further  are 
referred  to  his  writings. 

Mutual  Induction — -Transpositions. — If  a  three-phase  line 
is  arranged  so  that  each  wire  in  turn  occupies  the  central 
position  over  a  distance  equal  to  one-third  of  the  entire 
length  of  line,  it  should  have  no  inductive  effect  on 
neighbouring  parallel  conductors.  The  influence  of  one 
power  circuit  on  a  neighbouring  power  circuit,  even  if 
carried  on  the  same  set  of  poles,  is  usually  negligible; 
and  the  transposition  or  "spiralling"  of  high-tension 
wires  is  generally  avoided  in  modern  installations. 
Trouble  is  more  likely  to  occur  with  telephone  wires. 
These  are  preferably  carried  on  a  separate  set  of  poles 


POWER   TRANSMISSION 


as  far  away  as  possible  from  the  high-tension  line  ;  and, 
whether  the  power  wires  are  transposed  or  not,  it  is  well 
to  transpose  the  telephone  wires  at  frequent  intervals  — 
if  possible,  at  every  pole.  A  good  quality  of  insulator 
should  be  used  for  supporting  the  telephone  wires. 


CHAPTER  VII 

POLYPHASE    INDUCTION    MOTORS 

72.  THE  leading  principles  underlying  the  rotation  of 
a  short-circuited  armature  in  a  rotary  magnetic  field, 
produced  by  two  or  more  alternating  currents  differing 
in  phase,  were  explained  in  article  31,  Chapter  III., 
to  which  the  reader  is  referred,  and  with  which  he 
should  make  himself  thoroughly  familiar.  A  picture  of 
the  rotating  field,  dragging  with  it  the  short-circuited 
armature,  is  a  very  useful  one ;  but  there  is  another 
way  of  studying  the  behaviour  of  the  induction  motor 
which  will  generally  be  found  to  have  advantages.  This 
consists  in  considering  the  induction  motor  as  a  slightly 
modified  transformer,  the  primary  coils  of  which  are 
represented  by  the  stator,  or  inducing  windings,  while 
the  secondary  coils  are  movable,  being,  in  fact,  the  short- 
circuited  windings  of  the  rotor.* 

It  has  been  shown  (see  article  29,  Chapter  III.)  how 

*  This  nomenclature  is,  of  course,  based  on  the  assumption  that 
the  primary  windings — i.e. ,  the  coils  connected  to  the  supply  mains 
— are  at  rest,  while  the  short-circuited  secondary  revolves  ;  but, 
although  this  is  the  most  usual  arrangement,  it  must  be  understood 
that  it  is  by  no  means  a  necessary  one.  If  the  primary  coils  are 
wound  on  the  revolving  portion  of  the  magnetic  circuit,  then  slip 
rings  must  be  provided  for  conveying  the  current  to  the  coils, 
corresponding  in -number  to  the  number  of  phases  of  the  polyphase 
supply. 

193  13 


194  POLYPHASE   INDUCTION    MOTORS 

a  rotary  field  may  be  produced  by  two  or  more  currents 
having  certain  phase  differences  between  them ;  and 
since  two  equal  currents  differing  in  phase  by  go  time- 
degrees  will  produce  a  uniform  rotating  field,  it  follows 
that  any  uniform  rotating  field,  however  produced,  can 
be  resolved  into  two  equal  alternating  components  differ- 
ing in  phase  by  a  quarter  period ;  and,  instead  of  study- 
ing the  problems  of  induction  motors  by  picturing  a 
magnetic  field  revolving  in  space,  we  shall,  in  this 
chapter,  treat  the  subject  by  imagining  two  alternating 
magnetic  fields  differing  in  phase  by  one  quarter  of 
a  period,  and  acting  together  so  as  to  bring  about  the 
rotation  of  the  short-circuited  armature. 

This  method  is  by  no  means  new  :  indeed,  it  has 
always  been  advocated  by  Mr.  LI.  B.  Atkinson,  and  was 
used  by  him  so  long  ago  as  1898,  in  his  classic  paper  on 
"  Alternating-Current  Motors,"  read  before  the  Institution 
of  Civil  Engineers/1' 

Mr.  Atkinson,  however,  in  common  with  many  later 
writers,  treats  the  subject  very  fully,  and  even  goes  into 
details  of  design ;  while  the  methods  and  diagrams 
adopted  in  the  following  pages,  although  not  primarily 
intended  for  the  use  of  designers,  are  yet  sufficiently 
accurate  for  most  practical  purposes,  and  lead  to  some 
simplifications. 

In  Chapter  V.  the  theory  of  the  alternating-current 
transformer  was  explained  on  the  assumption  that  mag- 
netic leakage  was  nil — i.e.,  that  all  the  magnetism 
generated  in  the  primary  circuit  passed  also  through 
the  secondary  coils.  But  the  alternating- current  in- 
duction motor — owing  to  the  disposition  of  the  windings 
and  the  necessary  air-gap — is  by  no  means  a  perfect 
transformer,  and  we  shall  therefore  consider,  in  the  first 
*  Minutes  of  Proceedings,  Inst.  C.  E.,  vol. 'cxxxiii.,  p.  113. 


MAGNETIC   LEAKAGE   IN   TRANSFORMERS      195 

place,  what  are  the  effects  of  magnetic  leakage  in   an 
ordinary  single-phase  transformer. 

73.  Magnetic  Leakage  in  Transformers. — Unless 
the  primary  and  secondary  coils  of  a  transformer  are 
wound  close  together  on  the  same  portion  of  the  iron 
core,  there  must  necessarily  be  a  certain  number  of 
magnetic  lines  generated  by  the  primary  current  which 
do  not  thread  their  way  through  all  the  turns  of  the 


FIG.  78. 

secondary  coils.  The  amount  of  this  leakage  magnetism 
will  increase  with  the  growth  of  the  primary  current, 
and  will,  therefore,  be  approximately  proportional  to  the 
value  of  the  secondary  current. 

Fig.  78  is  a  section  through  a  single-phase  trans- 
former in  which  the  magnetic  leakage  would  be  consider- 
able. The  two  primary  inducing  coils,  marked  P,  are 
wound  upon  the  iron  core  at  some  distance  from  the 


196  POLYPHASE  INDUCTION   MOTORS 

secondary  coil,  S,  and,  moreover,  an  air-gap  has  been 
introduced  in  that  portion  of  the  magnetic  circuit  which 
lies  between  the  two  sets  of  coils. 

When  there  is  no  current  in  the  secondary — i.e.,  when 
this  is  open-circuited — nearly  all  the  magnetic  lines  due 
to  the  primary  current  will  pass  through  the  central 
core,  and  so  thread  their  way  through  the  secondary 
coil,  because — unless  the  air-gap  be  very  large— this  will 
be  the  easier  path  ;  and  the  amount  of  magnetism  rind- 
ing a  shorter  path,  such  as  «',  will  be  small.  But  when 
current  is  allowed  to  flow  in  the  secondary  coils,  this 
produces  a  magnetising  force  exactly  equal  and  opposite 
to  the  magnetising  force  of  the  main  component  of  the 
current  in  the  primary  coil.  The  result  is  that,  although 
this  component  of  the  primary  current  can  produce  no 
magnetic  flux  through  the  secondary  coil,  it  will  give 
rise  to  an  appreciable  amount  of  magnetism  which  will 
leak  across  the  air  spaces  and  miss  the  secondary  coils. 

In  treating  of  magnetic  leakage  in  transformers  and 
induction  motors,  the  assumption  is  usually  made  that 
the  secondary  winding  has  self-induction  ;  the  argument 
being  as  follows : 

The  current  in  the  primary  coils  gives  rise  to  a  total 
flux,  N,  of  such  a  value  as  to  choke  back  the  whole  of 
the  applied  potential  difference  which  is  not  lost  in  ohmic 
drop.  Of  this  total  flux,  Nf  lines  pass  through  the 
secondary  coils,  and  N/  lines  leak  back  by  other  paths. 
Now,  the  Nj  lines  through  the  secondary  induce  a 
certain  back  E.M.F.  which,  however,  is  not  in  phase 
with  the  secondary  current,  even  if  the  terminals  are 
short-circuited  or  the  outside  load  is  non-inductive.  It 
is  assumed  that  the  secondary  windings  have  self- 
induction,  and  that  the  current  flowing  in  them  gives 
rise  to  another  E.M.F.,  in  quadrature  with  the  E.M.F. 


MAGNETIC   LEAKAGE   IN   TRANSFORMERS      IQ7 

due  to  the  flux  Nj,  thus  reducing  the  useful  E.M.F. 
producing  the  flow  of  current. 

It  is  not  suggested  that  this  is  an  incorrect  way  of 
treating  the  subject  (in  some  cases  it  may  be  a  necessary 
one) ;  but  note  that  it  involves  the  idea  of  a  certain 
portion  of  the  total  leakage  magnetism  being  generated 
by  the  current  in  the  secondary  coil;  and  we  must, 
therefore,  conceive  of  two  independent  streams  of  alter- 
nating magnetism  (with  a  phase  difference  of  a  quarter 
period)  passing  through  this  coil.  In  other  words, 
the  magnetic  flux  due  to  the  primary  coil  is  considered 
as  giving  rise  to  a  certain  E.M.F.  in  the  secondary  coil, 
the  current  in  which  produces  a  back  E.M.F.  of  self- 
induction,  which,  together  with  the  above  E.M.F.  of 
mutual  induction,  gives  us  the  resultant  useful  E.M.F.  in 
the  secondary  winding. 

The  writer's  method,  which  leads  to  a  certain  simpli- 
fication of  the  diagrams,  is  based  on  the  assumption 
that  the  whole  of  the  magnetic  flux  which  passes 
through  the  secondary  coil  generates  an  E.M.F.  in  the 
secondary  windings  to  which  the  secondary  current  is 
directly  due,  and  that  this  is  the  only  E.M.F.  generated  in 
these  windings. 

Referring  again  to  Fig.  78,  it  might  be  objected  that 
the  current  in  the  secondary  coil,  S,  is  bound  to  produce 
leakage  magnetism  as  represented  by  the  dotted  line  a  b  ; 
but,  on  consideration,  it  will  be  seen  that  any  such  mag- 
netic lines  may,  with  equal  correctness,  be  associated 
with  the  component  of  the  primary  current  which 
balances  the  secondary  current ;  in  which  case  the  closed 
path  of  such  lines  will  not  be  through  the  central  iron 
core  on  which  the  secondary  is  wound,  but  through  the 
outside  portion  of  the  magnetic  circuit.  It  is  unques- 
tionable that  the  configuration  of  the  magnetic  circuit 


198  POLYPHASE    INDUCTION    MOTORS 

may  be  such  as  to  render  scientifically  inaccurate  this 
method  of  treating  the  subject ;  but,  in  almost  every  case 
which  is  likely  to  arise  in  practice,  the  error  introduced 
by  supposing  the  secondary  to  be  non-inductive  will  be 
negligible.  The  question  practically  resolves  itself  into 
the  correct  determination  of  the  core  loses  in  the  various 
portions  of  the  magnetic  circuit. 

74.  The  Induction  Motor  considered  as  a 
Special  Case  of  the  Alternate -Current  Trans- 
former.— Fig.  79,  which  represents  a  polyphase  induction 
motor  in  its  simplest  form,  has  been  specially  drawn  to 
illustrate  the  resemblance  between  such  a  motor  and  an 
ordinary  transformer  with  a  large  amount  of  leakage. 

The  revolving  field  is  resolved  into  two  equal  com- 
ponents with  a  phase  difference  of  90  degrees  ;  the  two 
stator  coils  P  P  being  connected  to  phase  i,  while  the 
two  coils  O  O  are  connected  to  phase  2. 

In  practice,  the  motor  would  probably  have  more  than 
two  pairs  of  poles,  and,  moreover,  the  stator  or  primary 
windings  would  be  distributed  in  a  number  of  slots  on 
the  inside  periphery  of  the  stator  ring ;  but  the  arrange- 
ment shown  in  the  figure  is  otherwise  correct,  and  will 
serve  the  purpose  of  explanation  better  than  a  section 
showing  the  actual  disposition  of  the  windings  in  a  two- 
phase  motor. 

The  rotor  windings  may  consist  of  several  equally 
spaced  short-circuited  windings  ;  or — if  the  squirrel-cage 
type  is  used — all  the  conductors,  passing  through  slots 
on  the  periphery  of  the  rotor,  would  be  short-circuited 
by  a  couple  of  rings  at  the  ends  of  the  armature.  In 
Fig.  79,  each  pair  of  diametrically  opposed  conductors  is 
supposed  to  be  short-circuited  by  end  connectors,  such 
as  n  and  m. 


ELEMENTARY   INDUCTION    MOTOR 


199 


The  action  of  the  motor  may  briefly  be  described  as 
follows : 

The  coils  P  P  of  phase  i  produce  an  alternating 
magnetic  field  in  the  direction  B  B'.  This  induces  no 
E.M.F.  in  the  rotor  coil  «,  of  which  the  axis  is  at  right 

|B 


A 


FIG.  79. 

angles  to  the  direction  of  the  magnetic  flux ;  but  in  all 
the  other  coils  E.M.F.s  will  be  generated  by  the  alter- 
nating flux,  and,  since  the  rotor  windings  are  assumed 
non-inductive,  the  resulting  currents  will  be  in  phase 
with  the  induced  E.M.F.,  and  therefore  exactly  one 


20O  POLYPHASE    INDUCTION    MOTORS 

quarter  period  behind  the  phase  of  the  alternating 
magnetism.  The  amount  of  current  flowing  in  each  coil 
will  be  proportional  to  the  total  flux  passing  through  it, 
being  greatest  in  the  coil  m  lying  on  the  axis  A  A'. 

If  the  motor  is  at  rest,  no  torque  will  be  exerted  by  these 
currents  reacting  upon  the  field  of  phase  I,  and  the  motor 
will,  therefore,  not  start  with  only  the  coils  P  P  excited, 
even  under  light  load  conditions.  Observe,  however, 
how  the  coils  O  O  excited  by  phase  2  produce  a  field 
along  the  axis  A  A7  exactly  90  time-degrees  out  of  phase 
with  the  field  B  B',  and,  therefore,  exactly  in  phase  with 
the  currents  induced  by  phase  i  in  the  rotor  windings. 

The  result  is  that  the  magnetism  entering  the  rotor 
from  one  set  of  poles  gives  rise  to  currents  in  the  rotor 
conductors  in  phase  with  the  magnetic  flux  from  the 
other  set  of  poles  ;  and  since  the  torque  will  depend 
upon  the  product  of  the  current  and  field  strength — i.e., 
upon  the  magnitude  of  the  rotor  currents  and  the  in- 
tensity of  that  component  of  the  magnetic  field  in  which 
the  rotor  conductors  are  immersed  which  is  in  phase 
with  the  current — it  follows  that  the  motor  will  be  self- 
starting. 

As  the  speed  increases,  the  E.M.F.s  induced  in  the 
rotor  conductors,  due  to  rotation  in  the  two  alternating 
magnetic  fields,  will  tend  to  reduce  the  amount  of  the 
rotor  currents  ;  and  the  maximum  speed  will  be  attained 
when  the  E.M.F.s  generated  in  the  conductors,  as  they 
cut  through  the  magnetic  flux,  almost  exactly  equal  and 
balance  the  E.M.F.s  due  to  induction — the  difference 
being  such  as  to  allow  the  very  small  current  to  pass 
which  is  necessary  to  overcome  bearing  friction,  iron 
loss,  windage,  etc. 

The  maximum  possible  speed,  if  there  were  no  losses 
in  the  motor,  would  be  the  speed  of  synchronism;  and, 


ELEMENTARY    INDUCTION    MOTOR  2OI 

since  the  percentage  losses,  when  running  light,  are  very 
small,  an  unloaded  induction  motor  may  be  said  to  run 
practically  at  synchronous  speed.  In  order  to  illustrate 
and  amplify  this  statement,  it  should  be  mentioned  that 
by  synchronous  speed  is  meant  a  speed  of  rotation  equal 
to  that  of  the  revolving  field.  If  the  primary  coils  of  a 
machine  are  so  wound  as  to  produce  only  two  poles  per 
phase,  as  in  Fig.  79,  the  magnetic  field  will  revolve  at 
the  rate  of  /  revolutions  per  second,  all  as  explained  in 
Chapter  III.,  article  29.  If  there  are  four  poles  per 
phase,  the  rotating  field  will  make  only  //2  revolutions 
per  second,  and  so  on,  the  synchronous  speed  being  ex- 
pressed by  the  ratio  f/p,  where  p  is  the  number  of  pairs 
of  poles  per  phase,  and  /  is  the  frequency. 

Consider  a  rotor  coil  of  S  turns,  the  "  pitch  "  of  which 
is  the  angular  distance  between  consecutive  poles  of 
opposite  sign  on  one  phase.  The  average  value  of  the 
E.M.F.  induced  in  such  a  coil  by  the  alternating  flux  is 

4N/S, 

and  the  corresponding  value  of  the  E.M.F.  due  to  rota- 
tion is 

WX2/XNX2S, 

where  n  =  revolutions  per  second,  and  N  =  maximum  value 
of  the  flux  per  pole.  If,  now,  we  equate  these  two  quan- 
tities to  obtain  the  condition  of  maximum  possible  speed, 
we  get 


which  confirms  the  earlier  statement  that  the  maximum 
possible  speed  —  reached  only  if  the  losses  can  be  con- 
sidered negligible  —  is  the  speed  of  synchronism. 

Thus,  if  the  motor  be  supposed  to  have  only  one  pair 


202  POLYPHASE   INDUCTION    MOTORS 

of  poles  per  phase,  as  in  Fig.  79,  and  if  the  frequency  of 
the  stator  current  be  25,  then  the  number  of  revolutions 
per  minute  when  running  light  will  be  25  x  60,  or  1,500, 
because  this  is  the  speed  which  will  make  each  con- 
ductor cut  the  magnetic  flux  of  any  one  phase  in  both 
directions  during  the  time  of  one  complete  cycle  of 
magnetisation. 

75.  Vector  Diagram  for  Induction  Motor  with 

Rotor  at  Rest. — The  condition  of  things  at  the  moment 
of  switching  on  the  stator  current,  while  the  rotor  is  still 
at  rest,  is  represented  by  the  diagram  Fig.  80. 

The  construction  of  this  diagram  is  generally  similar 
to  that  of  Figs.  51  and  52  in  Chapter  V. ;  but  whereas 
these  refer  to  a  transformer  supposed  to  be  without 
magnetic  leakage,  Fig.  80  refers  to  a  transformer — such 
as  an  induction  motor — in  which  magnetic  leakage  plays 
an  important  part. 

If  the  rotor  is  of  the  squirrel-cage  type,  the  condition 
is  that  of  a  transformer  having  appreciable  magnetic 
leakage  and  a  short-circuited  secondary.  If,  on  the  other 
hand,  the  motor  under  consideration  has  a  wound  rotor 
with  resistances  connected  in  series  (for  reasons  which 
will  be  explained  later),  then  the  condition  is  merely  that 
of  a  transformer  with  its  secondary  closed  on  a  definite 
non-inductive  load.  The  diagram  Fig.  80  applies  to 
either  case,  and,  for  the  purpose  of  simplifying  the  con- 
struction, we  shall  assume  (as  when  treating  of  static 
transformers)  that  the  ratio  of  primary  to  secondary 
turns  is  i  :  i. 

It  must,  moreover,  be  clearly  understood  that  the 
diagram  represents  the  relation  of  E.M.F.  and  current 
in  only  one  of  the  phases,  such  as  that  which  excites  the 
coils  P  P  (Fig.  79) ;  but  exactly  the  same  relation  will 
hold  good  in  the  other  phase  (coils  O  O,  Fig.  79),  with 


INDUCTION    MOTOR   WITH    ROTOR   AT    REST      203 

the  one  exception  that  the  diagram,  as  a  whole,  must  be 
supposed  to  be  turned  through  an  angle  of  90  degrees. 

The  construction  of  the  vector  diagram  Fig.  80  is  as 
follows : 

Draw  the  vertical  line,  B'  B,  to  represent  the  phase 
of  that  component  of  the  total  magnetic  flux  which  passes 


A 


A 


FIG.  80. 

through  the  rotor.  This  will  induce  an  E.M.F.  O  E2  in 
the  rotor  windings,  lagging  exactly  a  quarter  period 
behind  the  magnetism,  O  B  ;  the  magnetising  component 
of  the  primary  current  being  O  !„„  in  phase  with  the 
magnetism.  The  rotor  current  O  I2  will  be  in  phase 
with  E2,  and  equal  in  magnitude  to  E2-i-R,  where  R  is 


204  POLYPHASE   INDUCTION    MOTORS 

the  resistance  of  the  rotor  windings.  (It  must  not  be 
forgotten  that  the  secondary  or  rotor  windings  are  sup- 
posed to  be  without  self-induction.) 

This  secondary  current  must  be  balanced  by  a  pri- 
mary current  component,  O  lv  exactly  equal  but  opposite 
to  O  I2  ;  the  total  primary  current  being  O  I,  obtained 
by  compounding  O  It  and  O  Im* 

Of  the  two  components  of  this  total  primary  current, 
it  is  the  "  wattless  "  component,  O  lmt  which  produces  the 
flux  through  the  rotor,  while  the  component  O  Ij  gives 
rise  to  the  leakage  flux  which,  although  indirectly  due  to 
the  secondary  current,  does  not  pass  through  the  rotor. 
This  leakage  flux,  being  in  phase  with  O  lv  produces  an 
E.M.F.  component  O  E3  in  the  primary  windings  only, 
exactly  one  quarter  of  a  period  behind  O  Ir 

By  combining  this  E.M.F.  with  the  induced  E.M.F., 
O  E2 — due  to  that  portion  of  the  total  flux  which  passes 
through  the  rotor — we  obtain  the  vector  O  E',  which  is  the 
total  back  E.M.F.  in  the  primary  circuit.  If  we  assume 
the  resistance  of  the  primary  coils  to  be  negligible,  the 
impressed  primary  E.M.F.  will  be  O  E,  exactly  equal 
and  opposite  to  O  E' ;  the  power  factor,  for  this  particular 

*  We  are  neglecting  the  losses  in  the  motor,  which  would  make 
O  I  slightly  greater  than  as  obtained  by  this  construction  ;  but,  in 
any  case,  the  "  energy  "  component  of  the  total  magnetising  current 
is  very  small  in  a  motor  in  comparison  with  the  "  wattless  "  com- 
ponent, O  Im.  This  is  due  to  the  necessary  air-gap  ;  and,  whereas 
in  a  well-designed  static  transformer  the  true  magnetising  current 
might  be  about  45  degrees  in  advance  of  the  magnetic  flux  (see 
Figs.  51  and  52)  in  a  motor,  it  would  be  much  more  nearly  in  phase 
with  the  magnetism.  Thus,  when  the  motor  is  running  light,  at 
synchronous  speed,  the  "energy"  component  of  the  no-load 
primary  current  —  required  to  overcome  frictional,  windage, 
hysteresis,  and  eddy-current  losses — would  not  exceed  about  15  per 
cent,  of  the  "  wattless  "  component,  O  Im>  which  is  in  phase  with 
the  magnetism. 


STARTING   TORQUE   OF   INDUCTION    MOTOR      205 

condition  of  the  rotor  at  rest,  being  cos  0,  where  0  is  the 
angle  between  the  vectors  O  I  and  O  E. 

76.    Starting  Torque    of    Polyphase   Induction 

Motor. — Referring  still  to  the  diagram  Fig.  80,  we  see 
that  the  rotor  current  I2  is  exactly  a  quarter  period  out 
of  phase  with  the  magnetic  induction,  O  B,  which  enters 
the  rotor.  The  rotor  current  cannot,  therefore,  react 
upon  this  field  so  as  to  produce  a  starting  torque  ;  and  if 
one  phase  only  of  a  polyphase  motor  is  connected  to  the 
supply,  the  motor  will  not  be  self-starting.  But,  as  pre- 
viously pointed  out,  the  other  set  of  coils  (O  O,  Fig.  79) 
produces  a  magnetic  flux  through  the  rotor,  equal  in 
amount,  but  exactly  90  degrees  out  of  phase  with  the  flux, 
B  B',  due  to  the  coils  P  P.  This  flux,  being  in  the 
direction  A  A'  (Fig.  80),  is  exactly  in  phase  with  the  current 
O  1 2  in  the  rotor,  as  already  explained  on  p.  200. 

The  starting  torque  will,  therefore,  be  proportional  to 
rotor  current  x  magnetic  flux  entering  rotor,  or,  since  O  E2 
is — to  a  certain  scale — a  measure  of  the  magnetic  flux 
component  which  passes  through  the  rotor,  we  may  write 

starting  torque  <*  O  I2  x  O  E2. 

In  Fig.  80,  if  the  impressed  stator  E.M.F.  is  supposed 
constant,  the  ends  of  the  vectors  O  E'  and  O  E  must 
necessarily  move  on  the  dotted  circle  described  from  O 
as  a  centre,  and  it  will  be  an  easy  matter  to  study  the 
effect,  upon  the  starting  torque,  of  varying  the  resistance 
of  the  rotor  windings,  while  maintaining  constant  the 
supply  pressure  at  the  stator  terminals. 

As  regards  the  running  condition,  it  is  evidently  an 
advantage  to  keep  down  the  resistance  of  the  rotor  con- 
ductors in  order  to  avoid  large  PR  losses  :  but  note,  at 
the  outset,  that  if  the  rotor  conductors  were  entirely  without 
resistance,,  the  starting  torque  would  be  nil.  The  smallest 


206 


POLYPHASE   INDUCTION    MOTORS 


amount  of  magnetic  flux  passing  through  the  rotor  would 
give  rise  to  an  enormous  current  in  the  rotor  conductors ; 
the  result  being  that  the  current  I2  would  immediately 
rise  to  such  a  value  as  to  cause  the  whole  of  the  primary 


Resistance  of  Rotor  Windings 
FlG.  8l. 


flux  to  be  leakage  magnetism,  in  the  phase  O  A,  thus 
bringing  O  E'  to  coincide  with  O  B,  while  O  E2  would 
be  equal  to  zero. 

The  curves  of  Fig.  81  have  been  plotted  from  measure- 


STARTING   TORQUE   OF   INDUCTION    MOTOR      2O/ 

ments  taken  off  Fig.  80  :  the  upper  curve  shows  how  the 
starting  torque  reaches  a  maximum  for  a  definite  resist- 
ance, R,  of  the  rotor  windings,  after  which,  for  any  further 
increase  of  resistance,  the  torque  decreases  rapidly.  This 
particular  value  of  the  resistance  is  such  as  to  cause  the 
leakage  magnetic  flux  to  be  equal  to  the  useful  flux  which 
passes  through  the  rotor.  In  other  words,  it  is  the  rotor 
resistance  which  will  make  E3,  in  Fig.  80,  equal  to  E2, 
thus  causing  the  product  E2  x  E3  to  be  a  maximum. 
The  reason  of  this  is  not  far  to  seek,  for  the  induced 
leakage  E.M.F.,  O  E3,  may  be  taken  as  being  propor- 
tional to  the  current  producing  it  (seeing  that  the  path  of 
the  leakage  magnetism  is  largely  through  air,  which  is  of 
constant  permeability).  The  length  of  the  vector  O  E3 
is,  therefore,  a  certain  multiple  of  the  length  O  I2,  and, 
instead  of  writing 

torque  <x  O  E2  x  O  I2, 
we  may  write 

torque  a  O  E2  x  O  E3. 

With  regard  to  the  resistance,  R,  this  is  evidently  equal 
to  O  E2  -r-  O  I2,  which  can  also  be  written 

rotor  resistance  oc  O  E2  4-  O  E3. 

The  upper  curve  in  Fig.  81  is  obtained  by  plotting 
the  product  (O  E2  x  O  E3)  with  the  corresponding  ratio 
O  "P 

;  the  resultant,  O  E'  (or  O  E,  the  impressed  volts), 


being  supposed  of  constant  value. 

The  lower  curve  shows  the  relation  between  current 

and  resistance  ;  it  has  been  obtained  by  plotting  Q  -g2 

as  abscissae,  with  the  corresponding  values  of  O  E3  as 
ordinates. 

It  will  be  understood  that  both  curves  in  Fig.  81  will 


208  POLYPHASE   INDUCTION    MOTORS 

continue  to  approach  zero  value  as  the  resistance  of  the 
rotor  is  increased.  When  the  rotor  windings  are  open- 
circuited — i.e.,  when  R  =  infinity — the  torque  will  evi- 
dently be  zero,  since,  although  the  magnetic  field  through 
the  rotor  will  have  reached  its  maximum  value,  there 
will  be  no  current  to  react  upon  it  and  produce  a  torque. 

77.  Vector  Diagrams  for  Induction  Motor  with 
Armature  in  Motion. — When  the  starting  torque, 
acting  upon  the  armature  or  rotor,  causes  the  same  to 
revolve,  another  E.M.F.  is  generated  in  the  rotor  wind- 
ings. This  is  the  E.M.F.  of  rotation,  due  to  the  cutting 
of  the  alternating  magnetic  field  by  the  rotor  conductors  : 
its  value  will  depend  upon  the  amount  of  the  magnetic 
flux  cut  by  the  revolving  conductors,  and  the  velocity  at 
which  they  cut  through  this  field  ;  it  will,  moreover,  be 
in  phase  with  the  magnetism.  The  vector  diagram,  as 
drawn  in  Fig.  80,  must,  therefore,  be  modified  slightly, 
and  it  will  now  be  as  drawn  in  Fig.  82,  which  applies  to 
the  condition  of  a  motor  running  up  to  speed. 

Let  us  suppose  that  the  current  in  rotor — which  will 
be  less  than  at  starting,  on  account  of  the  back  E.M.F. 
of  rotation — has  been  reduced  to  the  value  represented 
by  the  length  of  the  vector  O  I2.  The  corresponding 
component  of  the  primary  current  will  be  O  I1?  which 
produces  the  leakage  field  in  the  direction  O  A.  This 
gives  rise  to  the  leakage  induced  E.M.F.  O  E3,  which 
will  be  approximately  proportional  to  O  lv  and,  therefore, 
of  a  smaller  value  than  in  the  diagram  Fig.  80.  This 
enables  us  to  obtain  the  vector  O  E2  as  representing  the 
induced  E.M.F.  in  rotor  windings — the  magnitude  of  the 
resultant  E.M.F.  vector,  O  E",  remaining  as  before. 

The  magnetising  current,  O  !,„,  which  has  to  produce 
an  increased  flux  through  the  rotor,  will  be  greater  than 
in  Fig.  86 ;  it  will  be  approximately  proportional  to 


INDUCTION  MOTOR  WITH  ARMATURE  IN  MOTION 


O  E2.  The  total  primary  current  will  be  O  I,  and  the 
impressed  E.M.F.,  O  E  —  drawn  equal  and  opposite  to 
O  E",  since  we  are  neglecting  the  I  R  drop  in  primary. 

Now  mark  off  O  Er,  in  the  same  phase  as  O  I2,  to 
represent  the  drop  of  pressure  in  rotor  windings.  (The 
length  O  Er  will  bear  the  same  proportion  to  O  I2  in 
Fig.  82  as  O  E2  bears  to  O  I2  in  Fig.  80.)  This  leaves 
Er  E2  to  be  balanced  by  an  exactly  equal  E.M.F.  of 


— -A 


rotation  due  to  the  condiictors  cutting  through  the  magnetic 
lines  which  enter  the  rotor  a  quarter  period  in  advance  of  the 
flux  O  B. 

The  action,  when  running,  may  be  summed  up  as 
follows : 

Phase  B  of  the  simple  two-phase  motor  indicated  by 
Fig.  79— when  connected  to  the  field  coils,  P  P — pro- 
duces a  magnetic  flux  in  the  direction  B'  B,  through  the 

14 


210  POLYPHASE   INDUCTION    MOTORS 

rotor.  This  gives  rise  to  currents  in  the  rotor  coils, 
which  are  in  phase  with  the  magnetic  flux  due  to  the  magnet- 
ising current  of  phase  A  in  the  field  coils  O  O.  When  the 
armature  begins  to  revolve,  a  back  E.M.F.  is  produced 
in  these  rotor  coils,  exactly  opposite  in  phase  to  the 
current  already  flowing.  It  tends,  therefore,  to  reduce 
this  current,  and  actually  does  so,  until  the  difference 
between  the  induced  E.M.F.  E2  (Fig.  82),  due  to  the 
magnetism  of  phase  B,  and  the  back  E.M.F.  due  to 
rotation  in  the  magnetic  field  of  phase  A,  leaves  a  re- 
sultant or  effective  E.M.F.,  Er,  just  sufficient  to  send 
such  a  current  through  the  rotor  as  will  give  the  neces- 
sary torque. 

It  will  be  understood  that  exactly  the  same  thing 
occurs  a  quarter  period  later,  when  the  same  coils  that 
occupied  a  horizontal  position  such  as  m  (Fig.  79)  now 
occupy  a  vertical  position  such  as  the  coil  n}  only  in  this 
case  the  field  due  to  phase  A,  exciting  the  coils  O  O, 
gives  rise  to  currents  in  the  rotor  coils,  while  the  back 
E.M.F.  of  rotation  in  these  coils  is  due  to  their  cutting 
through  the  magnetic  flux  from  the  poles,  P  P.  The 
arrangement,  in  fact,  is  symmetrical,  and  each  set  of 
stator  coils  gives  rise  to  currents  in  the  rotor  which  react 
upon,  and  are  in  turn  influenced  by,  the  magnetic  flux 
generated  by  the  other  set  of  coils. 

In  Fig.  83  the  diagram  has  been  drawn  for  the  con- 
dition of  practically  synchronous  speed — i.e.,  motor  run- 
ning light — the  only  work  done  being  that  required  to 
overcome  the  I2  R  losses  in  the  rotor.* 

The  whole  of  the  induced  E.M.F.  will  now  be  in 
the  direction  O  A7,  and  O  E2  will  be  equal  in  length  to 
the  vector  of  the  impressed  volts  O  E,  since  the  leakage 

*  For  simplicity  of  construction,  the  primary  losses  are  still  con- 
sidered negligible. 


INDUCTION  MOTOR  WITH  ARMATURE  IN  MOTION    2 1 1 

flux  due  to  the  very  small  rotor  current,  O  I2,  may  be 
neglected.  The  magnetising  current,  O  l^,  will  have 
reached  its  maximum  value,  and  the  total  primary 
current  will  be  O  I,  only  very  slightly  in  advance  of 
the  magnetising  component,  O  lm,  owing  to  the  com- 
paratively small  value  of  O  Ix.  In  a  well-designed  motor, 
the  total  primary  current,  O  I,  when  running  light,  might 
be  about  one  quarter  of  the  full-load  current. 


I 

E 

K 
\ 

3 

\ 

\ 

im     -.    -      y 

o,                          i 

It 

FIG.  83. 


Reverting  again  to  Fig.  82,  this  diagram,  if  studied 
attentively,  will  prove  very  instructive.  It  will  serve  to 
show  that  the  polyphase  motor  may  be  compared  with 
advantage  to  a  direct-current  shunt-wound  motor,  the 
behaviour  of  these  two  types  of  machines — both  when 
starting  up  and  when  running  under  load — being  in  many 
respects  similar. 

The  torque  exerted  by  the  armature,  as  explained  in 
article  76,  is  proportional  to  O  E2  x  O  I2 ;  and  since 


212  POLYPHASE   INDUCTION    MOTORS 

O  E3  oc  O  I2,  the  area  of  the  dotted  rectangle  O  E2  E"  E3 
is  a  measure  of  the  torque. 

Again,  the  E.M.F.  of  rotation,  represented  by  the 
length  E2  Er,  will  depend  upon  the  amount  of  the  flux 
entering  the  rotor,  and  the  rate  of  cutting,  or  speed  of 
the  motor ;  and  since  the  former  quantity  is  represented 
to  a  certain  scale  by  the  length  O  E2,  we  may  write 

length  E2  Er  oc  O  E2  x  speed, 
or,  speed  a  Q^- 

2 

With  regard  to  the  power  given  out  by  the  motor,  this 
is  the  product  of  torque  and  speed,  or 

power  oc  torque  x  speed 

=  (O  E2  x  O  I2)  x  ^^r 
=  O  I2  x  E2  En 

which  is  a  measure  of  the  power,  expressed  in  watts  per 
phase. 

This  is  graphically  represented  by  the  shaded  rect- 
angle (Fig.  82),  and,  since  the  conditions  which  occur 
in  phase  A  are  repeated  in  phase  B,  twice  the  watts 
obtained  by  the  above  construction  will  be  the  total 
power  developed  by  the  rotor. 

It  is  important  to  know  what  is  the  amount  of  overload 
that  an  induction  motor  will  stand — apart  from  consider- 
ations of  heating  or  efficiency — before  it  will  cease  to 
respond  to  a  further  increase  of  load;  and  this  point, 
together  with  the  reasons  for  inserting  resistances  in 
rotor  windings  for  starting  purposes,  will  be  dealt  with 
in  the  following  article. 


OVERLOAD   CAPACITY  OF   INDUCTION    MOTOR      213 

78.  Overload  Capacity  of  Induction  Motor  as 
influenced  by  Magnetic  Leakage  and  Rotor  Re- 
sistance.— Apart  from  considerations  of  mechanical 
strength  and  excessive  temperatures,  the  question  arises 
as  to  what  are  the  output  limitations  of  the  induction 
type  of  motor  on  momentary  overloads. 

The  output  is  the  product  torque  x  speed ;  but  as  the 
speed  of  an  induction  motor  varies  little  between  no 
load  and  full  load  conditions,  we  shall  consider,  in  the 
first  place,  the  production  of  a  maximum  torque  inde- 
pendently of  the  speed.  It  has  already  been  shown  that 

torque  oc  secondary  current  I2  x  secondary  induced  volts  E2, 

because  the  last  term  is  a  measure  of  the  magnetic  flux 
entering  the  rotor.  It  was  also  shown  that  the  vector 
O  E3  (Fig.  82),  representing  the  counter  E.M.F.  in  the 
primary  coils  due  to  the  leakage  flux,  is  approximately 
proportional  to  the  current  component  lv  and  therefore 
to  I2.  Thus,  in  terms  of  the  quantities  in  Fig.  82, 

torque  oc  O  E2  x  O  E3, 

and  this  product  will  have  a  maximum  value  when 
O  E2  =  O  E3  =  \/ 2  O  E.  This  leads  to  the  conclusion 
that  the  maximum  torque  will  be  obtained  when  the 
secondary  current  has  such  a  value  that  the  leakage  flux 
is  equal  to  the  useful  flux.  Also,  since  the  induced 
pressure  (E2),  which  corresponds  to  the  condition  of 
maximum  torque,  is  a  particular  multiple  of  the  im- 
pressed voltage  (E),  we  may  write, 

maximum  torque  oc  I'2  x  E, 
and,  for  a  constant  supply  voltage, 

maximum  torque  oc  I'2, 


214  POLYPHASE  INDUCTION   MOTORS 

where  I'2  is  the  particular  value  of  the  rotor  current 
which  makes  the  leakage  flux  equal  to  the  useful  flux. 
Obviously,  the  smaller  the  air-gap  between  stator  and 
rotor,  and  the  more  carefully  the  machine  is  designed 
with  a  view  to  keeping  down  the  leakage  magnetism,  the 
greater  will  be  the  required  value  of  I2  to  produce  the 
condition  of  maximum  torque.  It  is  the  magnetic  leakage 
due  to  the  rotor  current — or,  more  correctly  speaking, 
to  its  balancing  component  in  the  primary  circuit — which 
determines  the  maximum  value  of  this  current,  beyond 
which  any  further  increase  will  only  lead  to  a  reduced 
torque ;  and  since  the  motor  will  be  incapable  of  respond- 
ing to  a  further  addition  to  the  external  load,  it  will,  at 
this  point,  slow  down  and  come  to  rest. 

It  is  interesting  to  note  that  the  value  of  the  rotor  coil 
resistance  does  not  enter  into  these  considerations  of 
maximum  torque  ;  but  this  resistance,  since  it  involves 
the  idea  of  power  loss,  must  necessarily  enter  into  the 
determination  of  the  maximum  output. 

Referring  again  to  Fig.  82,  it  is  evident  that  the  length 
O  E2  will  be  constant  for  a  given  value  of  the  vector 
O  I2 ;  but  the  portion  O  Er  of  this  total  induced  E.M.F. 
in  the  rotor  will  depend  upon  the  resistance  of  the 
rotor  windings:  it  is — as  already  explained — the  E.M.F. 
required  to  overcome  the  resistance,  and  will,  therefore, 
be  equal  to  I2  x  R,  where  R  stands  for  the  rotor 
resistance. 

Hence,  for  a  given  secondary  current,  such  as  the 
critical  value  corresponding  to  the  condition  of  maximum 
torque,  the  length  O  Er  will  be  proportional  to  the  rotor 
resistance.  But  it  is  also  proportional  to  the  slip,  because 
— as  explained  on  p.  212 — the  rotor  speed  may  be 

"P    "P 

expressed  by  the  ratio      2     r,  which  is  the  same  thing 
U  .b2 


OVERLOAD  CAPACITY   OF   INDUCTION    MOTOR      215 

as  stating  that  the  length  E2  Er  is  a  measure  of  the  actual 
speed,  to  the  same  scale  as  O  E2  represents  the  maxi- 
mum or  synchronous  speed ;  thus  leaving  the  difference 
(O  E2  —  E2  Er)  or  O  Er,  to  stand  for  the  slip  revolutions, 
or  relative  speed  of  revolving  field  and  rotor. 

The  conclusion  to  be  drawn  from  the  foregoing  argu- 
ments is  that — although  the  maximum  torque  exerted  by 
a  motor  is  not  affected  by  the  resistance  of  the  rotor 
winding — the  drop  in  speed  which  must  occur  before  the 
maximum  torque  is  reached — i.e.,  before  the  machine  will 
cease  to  respond  to  a  further  increase  of  load— depends 
upon  the  rotor  resistance,  and,  indeed,  this  drop  in  speed, 
or  percentage  slip,  is  directly  proportional  to  the  resistance  of  the 
rotor  windings. 

Example. — If  the  particular  value  of  the  rotor  current 
producing  maximum  torque  is  100  amperes  per  phase, 
and  the  induced  E.M.F.  (E2)  per  phase  is  12  volts,  a  rotor 

resistance  of  —  =  -12  ohm  would  result  in  the  maximum 
100 

torque  being  exerted  at  the  moment  of  starting — i.e.,  when 
the  speed  =  o,  or  slip  =  100  per  cent.  But  if  we  reduce 
this  resistance  to  *o6  ohm,  the  maximum  torque  will  be 
exerted  when  the  speed  is  one-half  synchronous  speed 
(slip  =  50  per  cent.).  If  the  resistance  is  only  one-sixth 
of  the  above  value — i.e.,  '02  ohm — the  breakdown  speed 
would  be  five-sixths  of  synchronous  speed  (slip  =  one- 
sixth). 

The  curves  of  Fig.  84  have  been  drawn  to  illustrate 
this  point. 

Curve  A  would  apply  to  a  motor  without  any  external 
resistance  in  rotor ;  it  refers  to  a  machine  which  would 
break  down  when  the  slip  reaches  a  value  equal  to 
20  per  cent,  of  synchronous  speed,  or  five  times  what  we 
have  supposed  to  be  the  slip  at  normal  full  load  (4  per 


216 


POLYPHASE   INDUCTION    MOTORS 


cent.) :  the  maximum  torque  being — in  this  example — a 
little  over  three  times  the  full-load  torque. 


80 


100 


40  60 

Percentage  Slip 

FIG.  84. 

If  the  resistance  of  the  rotor  windings  is  increased  in 
the  proportion  of  5  to  2,  the  maximum  torque  will  occur 


OVERLOAD   CAPACITY   OF   INDUCTION    MOTOR      217 

at  half  the  synchronous  speed  (slip  =  50  per  cent.).  (See 
curve  B.) 

By  still  further  increasing  the  rotor  resistance,  until  it 
is  five  times  as  great  as  the  value  corresponding  to 
curve  A,  we  obtain  curve  C,  which  shows  how  the  torque 
would  increase  with  the  slip  until  the  maximum  value  is 
reached  with  the  rotor  at  rest  (slip  -  100  per  cent.). 

It  should  be  noted  that,  for  small  values  of  the  slip, 
the  curves  A,  B,  and  C  are  practically  straight  lines,  and, 
for  this  reason,  it  is  always  permissible  to  assume  that 
the  slip  is  directly  proportional  to  the  torque  for  all  values 
from  zero  up  to  the  rated  full  load  of  an  induction  motor. 

Although  it  is  the  maximum  torque  which  determines 
the  speed  at  which  the  motor  will  break  down — i.e.,  fail 
to  provide  the  increased  turning  effort  demanded  by  a 
further  increase  of  load — it  does  not  follow  that  this 
particular  speed  corresponds  with  the  greatest  possible 
output  or  brake  horse-power.  The  maximum  output 
would  occur  at  a  somewhat  smaller  slip  than  that  which 
corresponds  with  the  breaking-down  point  ;  because, 
although  the  torque  may  continue  to  increase,  the  speed 
is  being  reduced,  and  the  maximum  value  of  the  product 
speed  x  torque  will  occur  before  the  torque  has  reached 
its  maximum  possible  value. 

This  is  clearly  shown  by  the  dotted  curve  in  Fig.  84, 
which  has  been  plotted  to  show  the  relation  between  speed 
and  power  for  the  particular  rotor  resistance  corresponding 
to  the  curve  B.  It  will  be  seen  that  the  maximum  value 
of  this  dotted  curve  occurs  with  a  speed  of  75  per  cent, 
(slip  =  25  per  cent.),  whereas  the  maximum  value  of  the 
torque  curve  B  occurs  at  a  speed  of  50  per  cent,  of 
synchronism  (slip  =  50  per  cent.).* 

*  The  overload  capacity  of  a  polyphase  motor  just  before  falling 
out  of  step  would,  in  practice,  be  something  between  twice  to  three 
times  or  even  four  times  the  rated  maximum  load.  The  values  of 


2l8  POLYPHASE   INDUCTION    MOTORS 

79.  General  Conclusions  regarding  Magnetic 
Leakage  in  Induction  Motors. — The  objections  to  the 
leakage  magnetism,  which  is  produced  by  the  stator  coils, 
but  fails  to  pass  through  the  rotor,  are  fairly  evident. 
This  dissipation  of  magnetic  flux,  which  cannot  be 
utilised,  leads  to  a  reduction  in  the  overload  capacity 
of  the  motor,  and  a  diminished  torque,  for  a  given  slip, 
at  intermediate  loads.  It  also  reduces  the  power  factor, 
and  even  the  efficiency,  since  it  involves  increased 
hysteresis  and  eddy-current  losses  in  the  iron  cores  and 
neighbouring  metal  work. 

The  primary  magnetising  current  is  usually  between 
25  and  35  per  cent,  of  the  normal  full-load  current,  and  it 
should  clearly  be  the  aim  of  the  designer  to  keep  it  as  low 
as  possible. 

The  chief  remedy  for  leakage  troubles  in  alternating- 
current  motors,  and  other  alternating-current  apparatus, 
is  to  work  with  currents  of  low  periodicity ;  *  this  permits 
of  a  reduction  in  the  necessary  number  of  poles  in  the 
motor  for  a  given  speed  of  rotation,  and  also  allows  of 
higher  inductions  being  used  in  the  iron  cores.  This  is  a 
distinct  gain;  and  in  many  cases  where  improvements 
in  the  working  of  alternating-current  apparatus  are 
attributed  to  the  adoption  of  reduced  frequencies,  the 
improvements  are  not  to  be  accounted  for  by  the  reduction 
of  frequency  per  se,  but  to  the  fact  that  the  induction  in 
the  iron  cores  has  been  increased. 

slip  corresponding  to  curves  B  and  C  of  Fig.  84  are  larger  than 
would  occur  in  a  commercial  machine  :  the  slip  at  full  load  rarely 
exceeds  4  per  cent,  of  synchronous  speed. 

*  The  expense  of  working  at  very  low  frequencies  is,  however, 
great ;  especially  when  static  transformers  are  used— as  is  very 
often  the  case — for  raising  and  lowering  the  pressure  at  both  ends 
of  a  transmission  line. 


MAGNETIC   LEAKAGE    IN    INDUCTION    MOTORS      2IQ 

The  following  considerations  will  make  it  clear  that, 
with  a  view  to  keeping  down  the  percentage  leakage 
loss,  it  is  advantageous  to  make  the  magnetic  induction 
in  the  iron  as  high  as  possible ;  although  it  must  not  be 
overlooked  that,  for  a  given  frequency,  the  practical  limit 
in  this  direction  is  set  by  the  amount  of  the  hysteresis 
and  eddy-current  losses,  which  increase  largely  with  the 
higher  inductions :  but  by  keeping  down  the  frequency, 
higher  inductions  are  permissible. 

Let  Bs  stand  for  the  maximum  value  of  that  component 
of  the  total  induction  in  primary  core  which  passes  also 
through  the  secondary  or  rotor  windings,  and  B/  for  the 
maximum  value  of  the  leakage  component  of  the  total 
induction  in  primary  core. 

Then,  assuming  the  reluctance  of  the  path  of  the 
leakage  lines  to  be  constant,  we  may  write 

B/  oc  ampere-turns  tending  to  produce  leakage, 
oc  T  x  Ij, 

where  T  =  the  number  of  turns  in  the  primary  or  stator 
winding,  and  It  =  the  main  component  of  the  primary 
current.  And  since  the  mean  back  E.M.F.  in  primary, 
due  to  leakage  (the  vector  O  E3  in  the  last  few  diagrams), 
is  proportional  to  B/  x  T  x  /,  where  /  =  the  frequency, 
it  follows  that  we  may  write 

E3  or  IP  Ix  /, 

from  which  we  see  that  the  mean  value  of  the  "  leakage 
volts  "  for  any  given  arrangement  of  the  magnetic  circuit 
is  proportional  to  the  square  of  the  number  of  turns  in 
the  winding,  and  to  the  current,  and  to  the  frequency. 

But  as  it  is  not  the  actual  value  of  the  leakage  E.M.F. 
with  which  we  are  now  concerned,  it  will  be  advisable  to 
express  this  as  a  percentage  of  the  E.M.F.  available 


220  POLYPHASE   INDUCTION    MOTORS 

for  generating  current  in  the  secondary  circuit.  This  last 
will  be  proportional  to  B^  x  T  x  /;  and  if  we  divide  the 
actual  "  leakage  E.M.F."  by  this  amount,  we  obtain  the 
expression  : 

Percentage  back  E.M.F.  due  to  leakage 

T2  Lf 

re  1  / 

B,T/ 

x   Lk. 
B, 

It  follows  that,  if  we  consider  a  transformer  under  given 
load  conditions,  the  percentage  back  E.M.F.  due  to  leakage 
is  independent  of  the  frequency,  but  is  inversely  pro- 
portional to  the  induction  in  the  core  enclosed  by  the 
secondary  winding  ;  and  since  the  induction  motor  may 
be  regarded  as  a  transformer  with  short-circuited 
secondary,  the  above  reasoning  is  applicable  to  this 
special  form  of  alternating-current  transformer. 

80.  Complete  Vector  Diagram  for  Polyphase 
Induction  Motor.— Up  to  the  present  we  have  sup- 
posed the  stator  or  primary  winding  to  be  of  negligible 
ohmic  resistance,  and  it  will,  therefore,  be  advisable  to 
examine  briefly  how  the  problem  is  influenced  by  taking 
this  resistance  into  account. 

Fig.  85  is  the  complete  diagram  for  a  polyphase  induc- 
tion motor.  It  differs  in  only  two  respects  from  the 
diagram  Fig.  82.  In  the  first  place,  the  magnetising 
current  component,  O  Iw,  has  been  drawn  rather  more 
than  90  degrees  in  advance  of  the  induced  E.M.F.,  E2,  so 
as  to  take  into  account  the  small  "  energy  "  component 
of  the  total  primary  current,  required  to  provide  for  the 
hysteresis  and  eddy-current  losses.  In  the  second  place, 
another  component  O  E',,  of  the  primary  impressed 


DIAGRAM   FOR   POLYPHASE   INDUCTION    MOTOR    221 

E.M.F.  has  been  taken  into  account.  This  is  the  com- 
ponent required  to  overcome  the  resistance  of  the  stator 
windings. 

The  construction  of  the  diagram  Fig.  85  is  exactly 
similar  to  that  of  Fig.  82  up  to  the  point  of  obtaining  the 
primary  vector  O  Ex  (corresponding  to  O  E  of  Fig.  82)  ; 
but  we  have  now  to  add  another  component  O  E'r — 


FIG.  85. 

equal  to  I  x  R,  where  I  =  the  total  primary  current,  and 
R  =  the  primary  resistance — in  order  to  obtain  the 
required  potential  difference,  O  E,  at  stator  terminals. 
The  introduction  of  this  component  of  the  primary 
E.M.F.  leads  to  a  reduction  of  the  angle  0,  and,  there- 
fore, an  improvement  in  the  power  factor ;  but  there  is 
evidently  no  advantage  in  increasing  the  power  factor 
at  the  cost  of  additional  wasted  energy.  A  high  primary 


222 


POLYPHASE   INDUCTION    MOTORS 


resistance  cannot  be  otherwise  than  objectionable ;  it 
absorbs  a  certain  amount  of  energy  which  would  otherwise 
have  been  available  for  doing  useful  work. 

81.  Efficiency   of   Polyphase    Motors. — The   two 

shaded  parallelograms  in  Fig.  85  represent  respectively 
the  power  which  is  supplied  to  the  primary  terminals, 
and  that  which  is  transmitted  to  the  shaft  by  the  revolv- 
ing rotor. 

The  power,  per  phase,  supplied  to  the  primary  terminals 
is  E  x  I  x  cos  0,  and  it  is  proportional  to  the  area  of  the 
parallelogram  constructed  on  one  of  the  vectors,  O  E,  with 
the  other  vector,  O  I,  moved  round  through  an  angle  of 
90  degrees.  The  power  utilised  in  driving  the  shaft  is 
I2  x  E2  less  the  I2  R  losses  in  the  rotor  windings,  or 
1 2  x  (E2  —  Er),  and,  if  windage  and  friction  losses  are 
neglected,  the  efficiency  will  be  the  ratio  of  this  last  quantity 
to  the  power  put  into  the  motor  at  primary  terminals. 

The  full-load  efficiency  of  commercial  polyphase  motors 
will  vary  between  75  per  cent,  for  small  sizes  and  93  per 
cent,  for  large  sizes ;  and  it  will  fall  off  somewhat  on  an 
appreciable  overload. 

The  following  table,  compiled  from  figures  referring  to 
a  large  number  of  three-phase  motors  of  various  makes, 
may  be  of  use  as  indicating  the  average  full-load  efficiency 
and  power  factor  likely  to  be  obtained  with  this  class  of 
machine  : 


B.H.P. 

Efficiency. 

Power  Factor,  Full  Load. 

2 

•76 

•81 

5 

•81 

•84 

10 

•85 

•90 

20 

•88 

•91 

50 

•91 

•92 

20O 

'93 

•93 

MEASURING  THE  SLIP  OF  INDUCTION  MOTORS      223 

82.  Methods  of  Measuring-  the  Slip  of  Induc- 
tion Motors.  —  One  of  the  most  important  factors  to  be 
measured  when  testing  an  induction  motor  is  the  slip.  A 
large  slip  for  a  given  load  is  an  indication  of  large  losses 
in  the  rotor  windings.  We  have  already  seen  how  the 
slip,  for  a  given  torque,  is  proportional  to  the  resistance 
of  the  rotor  windings.  It  has  also  been  explained  in 
connection  with  the  vector  diagrams  (see,  for  instance, 
85)  that  the  slip,  expressed  as  a  fraction  of  the 


synchronous  speed,  is  ^  >  where  E,.  stands  for  the  volts 

required  to  overcome  rotor  resistance,  and  E2  is  the 
E.M.F.  induced  in  the  rotor  (the  difference,  E2  -  Er, 
being  the  back  E.M.F.  due  to  rotation  in  the  magnetic 
field). 

Now,  it  is  evidently  equally  correct  to  write 


percentage  slip  =  100  x 


*« 


which  amounts  to  defining  the  slip  as  the  ratio  of  rotor 
copper  losses  to  the  total  power  imparted  to  the  rotor 
conductors  by  induction  from  the  primary  circuit. 

If,  therefore,  we  know  the  rotor  resistance  and  current, 
we  can  calculate  the  slip.  If  the  rotor  is  of  the  squirrel- 
cage  type,  the  equivalent  resistance  can  be  ascertained  as 
follows  : 

Clamp  the  rotor  so  as  to  prevent  rotation  :  supply  a 
low  pressure  to  the  stator  terminals  until  the  required 
primary  current  is  obtained  :  measure  the  total  power 
supplied  to  the  primary,  and  deduct  the  calculated 
I2  R  losses  in  stator  windings.  This  leaves  a  quantity 
representing  with  sufficient  accuracy  the  I2  R  losses 
in  the  rotor,  corresponding  to  a  definite  primary  current. 


224  POLYPHASE   INDUCTION    MOTORS 

Then,  if  W  is  the  output,  expressed  in  watts,  the  slip  is 

P  R 


'  W+  PR- 

Evidently  the  slip  could  be  measured  by  taking  careful 
readings  of  the  speed  when  running  light  (synchronous 
speed),  and,  again,  when  running  under  load.  The 


FIG.  86. 

difference  would  be  the  slip :  but  as  this  is  a  very  small 
quantity  relatively  to  the  two  speed  measurements,  the 
inaccuracy  and  disadvantages  of  such  a  method  are 
obvious. 

A  number  of  satisfactory  methods  of  measuring  the 
slip  will  suggest  themselves  to  anyone  with  a  little 
ingenuity  who  cares  to  study  the  question.  Two  useful 
methods  will  be  briefly  referred  to. 


MEASURING  THE  SLIP  OF  INDUCTION  MOTORS      225 

Imagine  a  small  contact  stud  fixed  to  any  convenient 
point  on  the  rotor,  in  such  a  position  as  momentarily  to 
close  an  electric  circuit  once  during  each  revolution. 
This  circuit  should  be  connected,  through  a  voltmeter,  to 
the  same  source  of  supply  as  the  stator  windings  (see 
Fig.  86).  If  the  motor  were  running  at  synchronous 
speed,  the  indication  of  the  voltmeter  would  be  steadily 
of  the  same  value,  since  the  local  circuit  would  always  be 
closed  at  exactly  the  same  instant  in  the  cycle  of  the 
supply  pressure.  But  if  the  machine  is  running  at  some- 
thing less  than  synchronous  speed,  the  pointer  of  the 
voltmeter  will  take  up  a  swinging  motion.  This  is  due 
to  the  local  circuit  being  closed  each  consecutive  time 
slightly  later  in  the  E.M.F.  cycle ;  and  every  double 
oscillation  of  the  pointer  will  correspond  to  one  complete 
wave  of  alternating  E.M.F.,  thus  indicating  that — during 
the  time  of  one  such  double  swing — the  rotor  is  behind 
the  position  it  would  have  occupied  if  synchronous,  by 
twice  the  angle  subtended  by  two  consecutive  poles 
of  the  same  phase.  By  counting  the  number  of  double 
swings  per  second,  and  dividing  this  by  the  (known) 
frequency  of  supply,  we  get  the  percentage  slip. 

Methods  based  on  this  principle  have  given  very  satis- 
factory results.  Sometimes  a  telephone  receiver  is  used 
in  place  of  the  voltmeter,  with  advantage. 

Another  simple  method  of  measuring  the  slip  consists 
in  fixing  to  the  shaft  of  the  motor  a  circular  disc  having 
equally  spaced  black  and  white  segments  painted  upon  it, 
the  number  of  each  kind  corresponding  to  the  number  of 
pairs  of  poles  per  phase  in  the  stator. 

If  this  is  illuminated  by  an  arc  lamp  connected  to  the 
same  source  of  supply  as  the  stator,  the  disc  will  appear 
to  revolve  backward  ;  and  the  number  of  apparent  revo- 
lutions in  a  given  time  will  be  a  measure  of  the  slip, 

15 


226  POLYPHASE   INDUCTION    MOTORS 

Thus,  if  p  =-  the  number  of  pairs  of  poles  per  phase, 
and  /  =  the  frequency,  and  if  the  number  of  apparent 

P 

revolutions  be  counted  during  a  time  equal  to  100  x  -7 

seconds,  this  number  would  represent  the  percentage  slip. 
As  an  example,  suppose  p  =  8  and/  =  40.    Now  count 
the  number  of  apparent  revolutions  of  the  disc  during  a 
time  equal  to 

1,000  x   --  =  200  seconds 
4° 

=  3  minutes  20  seconds  ; 

let  us  say  that  38  apparent  revolutions  are  observed  in 
this  time ;  then  the  slip  corresponding  to  the  particular 
load  conditions  under  which  the  observation  was  made, 
would  be  3-8  per  cent* 

83.  Circle  Diagrams. — Although  the  vector  diagrams 
previously  used  are  useful  for  explaining  the  principles 
underlying  the  behaviour  of  induction  motors,  the  con- 
struction for  obtaining  the  primary  current  and  power 
factor  for  various  values  of  the  rotor  current  is  lengthy, 
and,  moreover,  in  the  case  of  Fig  85,  where  the  primary 
resistance  is  taken  into  account,  a  correction  has  to  be 
made  in  order  to  reduce  all  results  to  a  constant  value  of 
the  impressed  volts,  E.  It  is  true  that  this  correction 
merely  amounts  to  a  simple  proportion  sum,  determining 
the  scale  by  which  the  various  vectors  should  be  measured  ; 
but,  as  will  be  seen,  the  circle  diagrams  (originally  intro- 
duced by  Mr.  Alex.  Heyland)  have  much  to  recommend 
them,  and  are  very  convenient  to  use. 

*  For  a  description  of  an  ingenious  direct-reading  indicator  on 
the  above  principle,  see  article  by  Mr.  C.  V.  Drysdale  in  the 
Electrician  of  August  25,  1905. 


CIRCLE   DIAGRAMS 


227 


In  its  simplest  form,  the  circle  diagram  for  an  induc- 
tion motor  is  shown  in  Fig.  88.  This  applies  to  a  motor 
in  which  all  I2  R  losses  in  the  copper,  and  hysteresis  and 
eddy-current  losses  in  the  iron,  are  supposed  to  be  so 
small  as  to  be  negligible. 

The  corresponding  vector  diagram,  in  the  form  with 
which  we  are  already  familiar,  has  been  drawn  in 
Fig.  87. 


FIG.  87. 

It  will  be  seen  that  Fig.  87  is  practically  identical 
with  Fig.  82  (p.  209),  but  it  has  been  re-drawn  so  as  to 
facilitate  direct  comparison  with  the  circle  diagram. 

There  is  a  definite  relation  between  the  length  of 
the  rotor  current  vector,  O  I2,  and  the  magnetising 
current  component,  O  Im,  which  may  be  deduced  as 
follows : 

The  secondary  E.M.F.,  or  volts  induced  in  the  rotor, 
will  be  proportional  to  the  amount  of  the  magnetising 
current,*  or 

*  The  magnetic  circuit  is  supposed  to  be  of  constant  permeability, 
and,  therefore,  the  amount  of  the  magnetic  induction  is  proper- 


228  POLYPHASE   INDUCTION    MOTORS 

where  Km  is  a  constant.     Moreover,  the  leakage  volts, 
E3,  are  proportional  to  the  rotor  current,  I2,  or 

E3=I2xK2-  (a) 

where  K2  is  another  constant. 
A  third  condition  is  that 

E,'  +  E,«  =  E",  (3) 

where  E  stands  for  the  impressed  voltage,  and  is,  there- 
fore, a  constant. 

Inserting  values  for  E2  and  E3  from  equations  (i)  and 
(2),  we  get, 

T2        1^2         ,     T   2   I/"   2  _    T?2  /.\ 

1  nt   ^  m  +   X2     ^2     "      "*  •  (4j 

There  is  obviously  another  equation,  which  gives  the 
total  primary  current  in  terms  of  its  components,  and 
this  is, 

P-I».  +  V,  -    (5) 

because  the  triangle  O  I  lm  of  Fig.  87  is  right  angled, 
with  O  I  as  the  hypotenuse. 

Consider,  now,  the  circle  diagram  Fig.  88.  Draw 
the  vertical  line  O  E  to  represent  the  phase  of  applied 
E.M.F.,  and  on  O  B — at  right  angles  to  O  E — make 
O  A  equal  to  the  maximum  possible  value  of  the 
magnetising  current,  and  O  B  equal  to  the  maximum 
possible  value  of  the  rotor  current.  These  quantities 
are  expressed  in  terms  of  the  supply  voltage  and  the 
assumed  constants  thus : 


tional  to  the  current  producing  it.  This  assumption  is  justified  on 
account  of  the  comparatively  low  inductions  used  in  the  iron,  and 
the  relative  importance  of  the  air-gap  in  the  total  reluctance  of  the 
magnetic  circuit. 


CIRCLE  DIAGRAMS  229 

By  (i)  lm  =  E2  +  Km 

and  maximum  value  of  lm  =  O  A  =  E  -r  K;w ; 

also  by  (2)  I2  =  E3  -r  K2 

and  maximum  value  of  I2=OB  =  E-j-  K2. 

On  A  B  and  O  A  as  diameters,  describe  the  semi- 
circles as  shown.  Then,  for  all  intermediate  values 
of  the  primary  current,  the  end,  P,  of  the  primary 
current  vector,  O  P,  will  lie  on  the  larger  circle.  The 


FIG.  88. 

line  P  M  represents  the  rotor  current,  while  O  M  is 
a  measure  of  the  magnetising  component  of  the  total 
primary  current,  O  P.  The  angle  E  O  P  is  the  angle 
of  lag,  the  cosine  of  which  is  the  power  factor ;  and  the 
phase  relations  of  the  other  two  current  components 
are  also  correctly  shown  relatively  to  the  pressure 
vector,  O  E. 

This  diagram  correctly  reproduces  the  conditions  of 
Fig.  87,  although  in  a  somewhat  different  form.  That 
the  same  relations  still  hold  between  the  various  current 
vectors  can  be  proved  as  follows  ; 


230  POLYPHASE   INDUCTION    MOTORS 

In  the  first  place,  the  triangle  O  P  M  is  always  right 
angled,  with  O  P  as  the  hypotenuse,  and  this  is  all 
that  is  needed  to  satisfy  equation  (5).  In  regard  to  the 
other  relations  between  current  components,  the  triangle 
O  M  A  is  always  right  angled  whatever  may  be  the 
position  of  the  point  P  on  the  larger  semicircle.  Thus, 

(O  M)2  =  (O  A)2  -  (A  M)2, 

but  A  M  is  a  definite  fraction  of  P  M,  in  the  proportion 
of  O  A  to  O  B  ;  thus, 

AM  =  PM  x 


and  inserting  for  O  A  its  previously  ascertained  value 
in  terms  of  the  impressed  voltage,  we  get, 

F2        T  2K  2 
____    2      2 


which  value  for  the  square  of  the  magnetising  current  is 
the  same  as  that  given  by  equation  (4).  No  further 
proof  should  be  necessary  to  establish  the  complete 
similarity  between  the  circle  diagram  Fig.  88  and  the 
fundamental  vector  diagram  Fig.  87. 

It  is  interesting  to  note  that  the  maximum  possible 
power  factor  —  i.e.,  the  smallest  angle  6  —  occurs  when 
the  primary  current  vector  is  tangential  to  the  large 
semicircle,  as  indicated  by  the  dotted  line,  O  Pv  and 
this  shows  very  clearly  the  bad  effect,  as  regards  the 
power  factor,  of  a  large  air-gap  and  resulting  magnetising 
current,  O  A,  and  of  bad  design  generally,  leading  to 
a  large  leakage  'field,  or  increased  ratio  of  O  A  to  O  B. 
If  a  larger  air-gap  were  to  reduce  the  leakage  mag- 


CIRCLE  DIAGRAM  TAKING  LOSSES  INTO  ACCOUNT    23 1 

netism  due  to  the  rotor  current  proportionately  to  the 
increase  in  the  magnetising  current,  the  maximum 
possible  power  factor  would  not  alter;  but  this  is  not 
what  one  would  expect,  and,  indeed,  from  actual  tests 
made  by  Mr.  B.  A.  Behrend,  the  effect  of  enlarging  the 
air-gap  of  an  experimental  induction  motor  was  to 
increase  the  magnetising  current  in  the  ratio  of  3  to  7, 
while  the  short-circuited  rotor  current  (rotor  at  rest)  only 
increased  in  the  ratio  of  3  to  3-4  ;  thus  showing  the  im- 
portance of  keeping  the  air-gap  as  small  as  mechanical 
considerations  will  admit. 

84.  Circle  Diagram  taking-  Losses  into  Account. 

—If  we  take  into  account  the  iron  losses  and  the  resist- 
ance of  the  stator  coils,  the  diagram  Fig.  88  will  not  be 
strictly  correct ;  but  it  will  be  found  that  the  end,  P,  of 
the  current  vector,  O  P,  still  moves  upon  a  circle,  as  the 
load  on  the  motor  is  varied. 

Referring  to  the  complete  vector  diagram  Fig.  85 
(p.  221),  it  will  be  seen  that  the  impressed  E.M.F., 
E,  is  no  longer  exactly  equal  and  opposite  to  the  total 
back  E.M.F.,  E2;  but,  owing  to  the  resistance  of  the 
primary  winding,  it  is  greater  than  E2 ;  and  the  phase 
angle,  0,  between  total  current,  I,  and  impressed  volts, 
E,  is  smaller  than  it  would  be  if  the  primary  losses  were 
inappreciable.  The  other  point  of  difference  between 
Figs.  85  and  87  is  that  the  magnetising  component 
(O  lm)  of  the  primary  current,  is  now  more  than 
90  degrees  in  advance  of  the  rotor  current,  O  I2 ;  or,  in 
other  words,  the  angle  I  \m  O  in  the  triangle  O  I  \m  is 
something  greater  than  a  right  angle.  We  shall,  how- 
ever, assume  that  this  angle — i.e.,  the  phase  difference 
between  magnetising  and  rotor  currents — remains  con- 
stant under  all  conditions  of  load,  and  the  two  chief 
differences  between  the  circle  diagram  Fig.  88  and  the 


232  POLYPHASE    INDUCTION    MOTORS 

corrected    diagram    Fig.    89    may   be    summed    up    as 
follows  : 

i.  The  vector  of  the  no-load  magnetising  current  (O  A), 
in  the  corrected  diagram,  is  no  longer  at  right  angles  to 
the  pressure  vector  (O  E),  but  makes  an  angle  EGA 
with  the  vector  O  E,  which  is  less  than  90  degrees,  and 
corresponds  with  the  actual  measured  power  factor  when 
the  motor  is  running  light. 


FIG.  89. 

2.  The  angle  of  phase  difference  (O  M  P),  between 
rotor  current  and  magnetising  component  of  primary 
current,*  is  somewhat  greater  than  90  degrees,  and  will 
depend  upon  the  iron  losses  in  the  motor. 

The  diagram  Fig.  89  has  been  constructed  from 
measurements  taken  off  Fig.  85,  this  latter  diagram 
having  been  re-drawn  for  various  values  of  the  rotor 
current,  and  the  results  all  brought  to  a  common  basis 
of  comparison  by  supposing  the  impressed  volts  O  E  to 

*  Assumed  to  be  of  constant  value. 


CIRCLE  DIAGRAM  TAKING  LOSSES  INTO  ACCOUNT    233 

remain  of  a  constant  value.  It  will  be  seen  that  the 
centres,  C  and  D,  of  the  two  semicircles  no  longer  lie  on 
the  line  O  B,  at  right  angles  to  O  E,  but  above  this  line  ;* 
otherwise  the  diagram  is  very  similar  to  Fig.  88. 

The  maximum  possible  value  of  the  primary  current 
will  be  O  P',  which  corresponds  to  the  condition  of 
starting  (rotor  at  rest),  and  the  exact  position  of  the 
point  P'  on  the  circle  will  depend  upon  the  amount  of 
resistance  in  the  rotor  windings.  The  vector  O  P'  has 
been  drawn  on  the  assumption  of  a  reasonably  small 
rotor  resistance — which  would  result  in  a  loss  of  pressure 
somewhat  less  than  that  represented  by  the  length  O  Er 
in  Fig.  85 — but  with  an  increased  rotor  resistance,  the 
final  position  of  the  primary  current  vector  would  be  on 
the  circle  between  P'  and  P ;  as,  for  instance,  at  P". 

To  construct  such  a  diagram  which  shall  represent  the 
behaviour  of  an  actual  motor,  it  is  only  necessary  to  have 
the  same  particulars  as  would  be  required  for  the  con- 
struction of  a  vector  diagram  of  the  kind  drawn  in  Fig.  85 
(because  the  circle  diagram  is  merely  another  form  of 
this  general  diagram)  ;  these  particulars  are : 

1.  The  no-load  current  and  pressure  and  true  power. 

2.  The  current  and  pressure  and  true  power  with  rotor 
short-circuited,  and  at  rest. 

3.  The  resistance  of  the  primary  windings  per  phase. 
It  is  not  proposed  to  take  up  any  further  space  here  in 

order  to  explain  how  the  vector  diagram  may  be  drawn 
from  the  above  particulars  ;  but  if  the  reader  cares  to 
follow  the  matter  further,  he  is  referred  to  Appendix  II. 

*  In  Mr.  Heyland's  original  description  of  his  circle  diagram 
(see  £clairage  Electriqiie,  July  14,  1900),  he  makes  the  centre  of  the 
large  circle  fall  on  the  line  O  B,  at  right  angles  to  the  pressure 
vector  ;  but  this  will  not  be  found  to  give  results  in  strict  accord- 
ance with  the  data  obtained  from  actual  tests. 


234  POLYPHASE   INDUCTION    MOTORS 

at  the  end  of  the  book.  With  a  few  additional  lines,  the 
circle  diagram  can  be  made  to  give  practically  all  neces- 
sary information  regarding  the  performance  of  an  induc- 
tion motor  by  the  scaling  of  lengths,  with  occasional 
ratios,  or  products,  of  two  quantities.  Such  information 
would  include  speed,  slip,  torque,  power  losses,  and  effi- 
ciency, for  all  values  of  the  primary  current  vector  O  P. 
It  is  not  proposed  to  detract  from  the  simplicity  of  the 
diagram  Fig.  89  by  making  the  additions  referred  to ; 
but  before  leaving  the  subject  of  the  circle  diagram,  the 
main  features  of  Fig.  89  may  be  summed  up  as  follows  : 

O  E  represents  the  phase  of  the  impressed  E.M.F.  at 
primary  terminals.  The  other  straight  lines  are  current 
vectors  of  which  all  vertical  components  (parallel  to  O  E) 
are  "active"  currents,  while  all  horizontal  components 
(parallel  to  O  B)  are  "  wattless." 

O  P,  with  the  large  circle  as  the  locus  of  the  point  P, 
is  the  primary  current  vector. 

Cos  0  is  the  power  factor. 

P  M,  which  passes  through  the  junction  A  of  the  two 
semicircles,  is  the  secondary  current. 

O  M  is  the  magnetising  component  of  the  primary 
current. 

85.  Methods  of  starting  Induction  Motors. — To 

start  an  induction  motor  by  merely  closing  the  switches 
on  the  supply  circuit,  and  so  instantaneously  connecting 
the  terminals  to  the  full  supply  pressure,  is  evidently  un- 
satisfactory, on  account  of  the  very  large  rush  of  current 
which  would  occur ;  and,  if  the  motor  has  to  run  up  under 
load,  this  abnormally  large  current  might  continue  for  an 
appreciable  time. 

Such  a  method  of  starting  would  only  be  allowable  in 
the  case  of  very  small  motors,  although  it  is  sometimes 


METHODS   OF   STARTING   INDUCTION    MOTORS      235 

adopted  for  machines  as  large  as  10  b.h.p.  By  reducing 
the  pressure  across  terminals  at  the  moment  of  switching 
on,  through  the  introduction  of  resistances,  choking  coils, 
or  transformers  in  the  primary  circuit,  any  size  of  motor, 
with  short  circuited  rotor,  can  conveniently  be  started 
up,  provided  no  great  starting  torque  is  required ;  that  is  to 
say,  provided  the  motor  has  not  to  start  up  against  a 
heavy  load. 

The  method  generally  adopted  for  starting  motors  under 
load  consists  in  providing  the  motor  with  a  wound  rotor, 
the  three  sections  of  the  winding  having  their  starting 
ends  connected  at  a  common  junction,  while  their  finish- 
ing ends  are  taken  to  three  slip  rings,  from  which  the 
current  is  collected  by  brushes  in  the  usual  manner. 
These  brushes  are  connected  to  the  three  sections  of  a 
variable  non-inductive  resistance,  which  may  be  of  the 
liquid  type,  or  made  up  of  wire  spirals  joined  up  to  a 
simple  type  of  controller  having,  say,  half  a  dozen 
contacts  on  each  of  the  three  regulating  arms.  The 
other  ends  of  the  three  regulating  resistances  are  all 
permanently  connected  together. 

When  starting  up  the  motor,  the  controller  handle  is 
moved  to  the  position  where  the  three  rotor  circuits  are 
open — i.e.,  the  resistance  in  series  with  the  windings  is 
infinity.  The  stator  current  is  then  switched  on  under 
full  pressure,  and  the  controller  handle  moved  forward 
so  as  to  close  the  rotor  circuit  through  the  resistances, 
which  are  gradually  cut  out  as  the  machine  runs  up  to 
speed,  until,  when  full  speed  is  attained,  all  external 
resistance  is  cut  out,  and  the  rotor  windings  are  left  short- 
circuited  upon  themselves.  These  operations  are  equiv- 
alent to  inserting  a  starting  resistance  in  series  with  the 
armature  of  a  shunt- wound  direct-current  motor  ;  a  large 
starting  torque  can  be  obtained  in  this  manner,  as  will 


236  POLYPHASE   INDUCTION    MOTORS 

be  evident  if  the  reader  has  carefully  followed  article  76 
and  understood  the  meaning  of  the  curves  in  Fig.  81. 

It  has  been  shown  how,  by  suitably  increasing  the 
rotor  resistance,  the  torque,  with  rotor  at  rest,  may  be 
increased  up  to  the  maximum  torque  which  the  motor 
can  exert  under  any  conditions  of  working ;  whereas,  if 
the  resistance  of  the  short-circuited  rotor  is  small  (which 
it  should  be,  for  the  most  economical  running  conditions), 
the  torque  at  starting,  without  the  introduction  of  some 
external  resistance,  will  not  be  equal  to  the  maximum 
possible  torque,  although  the  current  taken  from  the 
supply  mains  will  be  greater  than  that  which  passes  when 
a  starting  resistance  is  inserted.  It  has  also  been  ex- 
plained how  this  effect  is  due  to  phase  displacement 
between  the  rotor  currents  and  the  magnetic  field  upon 
which  they  react ;  and  it  is,  therefore,  not  proposed  to 
dwell  any  longer  upon  this,  the  most  general  and  satis- 
factory method  of  starting. 

With  regard  to  the  other  method  already  referred  to 
— namely,  reducing  the  supply  pressure  at  starting  before 
connecting  the  motor  to  the  circuit — perhaps  the  most 
approved  way  of  effecting  this  is  by  means  of  auto-trans- 
formers. 

Fig.  90  shows  the  connections  to  a  three-phase  auto- 
transformer,  such  as  would  be  used  for  a  three-phase 
induction  motor  up  to  about  15  b.h.p.  size  ;  the  arrange- 
ment for  larger  motors  being  similar,  but  it  is  then 
advisable  to  arrange  for  two  or  more  tappings  from  each 
of  the  transformer  windings,  so  that  the  pressure  may  be 
increased  gradually,  instead  of  throwing  over  suddenly 
from  the  reduced  starting  pressure  to  the  full  line  pres- 
sure. 

As  shown  in  Fig.  go,  the  auto-transformer  consists  of 
the  usual  magnetic  circuit  of  laminated  iron ;  but,  instead 


METHODS   OF   STARTING   INDUCTION    MOTORS      237 

of  there  being  two  distinct  sets  of  coils  as  in  the  ordinary 
transformer,  each  limb  carries  only  a  single  continuous 
winding,  with  a  connection  tapped  off  at  a  suitable  inter- 
mediate point.  By  carefully  following  the  connections 
on  the  diagram,  it  will  be  seen  that,  when  the  three  poles 

Connections  from  Generator 


t 


Running 
Position 


Starting 
Position 


Connections  to  Stator  Windings 

FIG.  90. 

of  the  starting  switch  are  closed  on  the  lower  set  of 
contacts,  the  three  transformer  windings  have  their  free 
ends  connected  to  a  common  "  neutral "  bar ;  and  they 
will  consequently  act  merely  as  choking  coils,  or  as  the 
primaries  of  a  three-phase  transformer  with  open  secondary 
circuit.  It  will  be  noticed,  however,  that  the  tappings 


238  POLYPHASE   INDUCTION   MOTORS 

from  somewhere  about  the  centre  of  the  windings  go 
directly  to  the  terminals  of  the  motor,  which,  instead  of 
receiving  the  full  line  pressure,  get  only  a  reduced  pres- 
sure, depending  upon  the  ratio  of  the  number  of  turns 
comprised  between  the  points  A  and  B,  and  the  total 
number  of  turns  on  each  limb  of  the  magnetic  circuit. 
When  the  rotor  has  attained  a  fairly  high  speed,  the 
switch  is  thrown  over  to  the  upper,  or  running  position, 
thus  short-circuiting  the  portion  A  B  of  the  winding, 
and  leaving  the  motor  terminals  connected  directly  to 
the  supply. 

It  has  already  been  shown  how,  if  a  large  starting 
torque  is  required,  it  is  necessary  to  increase  the  rotor 
resistance  beyond  the  value  which  would  prove  econom- 
ical, or  even  possible,  for  continuous  running.  Many 
ingenious  methods  have  been  suggested  for  accomplish- 
ing this  end  without  the  use  of  sliding  contacts  and 
external  resistance  coils;  but  nearly  all  the  methods 
suggested  are  costly,  even  if  they  do  not  lead  to  un- 
desirable complications.  The  induction  motor  with 
1  'squirrel- cage  "  rotor  is  undoubtedly  the  simplest  and 
the  least  likely  to  give  trouble  when  entrusted  to  un- 
skilled hands,  and  the  method  of  starting  up  such  a 
machine  by  merely  closing  the  switch  connecting  the 
stator  windings  to  the  supply  has  much  to  recommend 
it.  It  is  possible  to  start  comparatively  large  motors  in 
this  way,  if  the  abnormal  rush  of  current  when  switching 
on  is  not  too  serious  an  objection  :  moreover,  for  certain 
conditions,  where  the  highest  efficiency  is  not  important, 
and  where  heating  troubles  are  not  likely  to  arise,  it  is 
possible  deliberately  to  design  the  rotor  with  a  com- 
paratively high  resistance — this  being  usually  provided 
in  the  end  rings  to  which  the  copper  conductors  are 
connected. 


SPEED   REGULATION  239 

In  conclusion,  it  has  been  shown  in  article  79  that,  by 
keeping  up  the  induction  in  the  iron,  the  starting  torque, 
for  a  given  arrangement  of  the  windings,  will  be  greater 
than  with  low  inductions.  If  the  best  starting  results 
are  to  be  obtained,  particular  attention  must  be  paid  to 
the  question  of  magnetic  leakage  (which  must  be  kept 
as  small  as  possible),  and  the  resistance  of  the  stator 
windings  must  be  small,  otherwise  the  effective  E.M.F. 
which  determines  the  amount  of  the  magnetic  flux  may 
be  considerably  less  than  the  potential  difference  at 
stator  terminals. 

86.  Speed  Regulation. — The  analogy  between  the 
induction  motor  and  the  shunt-wound  direct -current 
motor,  has  already  been  pointed  out ;  both  are  essen- 
tially constant- speed  machines,  and  it  is  usually  un- 
economical to  use  them  for  variable-speed  work.  A 
reduction  of  the  supply  voltage  does  not  meet  the  case, 
because,  although  the  required  back  E.M.F.  is  smaller 
than  with  the  normal  supply  voltage  across  terminals, 
the  magnetic  flux  through  the  armature  is  also  reduced, 
and  approximately  the  same  speed  is  required  to  produce 
the  (lower)  back  E.M.F. 

There  are  two  ways  of  regulating  the  speed  of  a  shunt- 
wound  direct-current  motor:  (i)  Inserting  resistance  in 
series  with  the  armature  winding,  while  the  full  voltage 
is  maintained  at  the  terminals  of  the  field  windings  ; 
(2)  inserting  resistance  in  series  with  the  field  windings, 
while  the  normal  supply  voltage  is  maintained  across  the 
brushes.  By  method  (i)  the  magnetic  induction  re- 
mains constant,  but  the  necessary  back  E.M.F.  will 
depend  upon  the  value  of  the  resistance  in  series  with 
the  armature.  If  this  is  such  as  to  absorb,  say,  25  per 
cent,  of  the  total  supply  volts,  the  required  back  E.M.F. 
will  now  be  75  per  cent,  of  what  it  would  be  under  the 


240  POLYPHASE   INDUCTION    MOTORS 

full  voltage ;  the  result  being  that  the  machine  will  run 
at  three-quarters  of  the  normal  speed.  This  method  is 
evidently  wasteful,  since  the  resistance  must  necessarily 
dissipate  the  whole  of  the  power  which  is  not  given  out 
by  the  motor  through  the  shaft. 

By  method  (2)  the  magnetic  induction  through  the 
armature  is  varied,  but  the  required  back  E.M.F.  re- 
mains constant.  The  speed  will  therefore  vary  inversely 
as  the  strength  of  the  magnetic  field.  This  method  is 
evidently  less  wasteful  than  the  first ;  but,  for  consider- 
able speed  variations,  difficulties  arise  owing  to  the  main 
field  being  necessarily  weak  for  the  higher  speeds,  while 
the  leakage  field  due  to  the  armature  currents  is  rela- 
tively large. 

Let  us  now  look  at  the  induction  motor.  Method 
(i)  is  almost  exactly  reproduced  by  inserting  resistance 
in  the  rotor  windings  ;  while  the  variation  in  the  induc- 
tion by  method  (2)  would  be  effected  by  altering  the 
frequency  of  the  supply.  (It  is  the  frequency  that — for  a 
given  impressed  E.M.F. — determines  the  value  of  the 
induction ;  and  the  speed  of  the  rotor,  when  running 
light,  will  therefore  vary  directly  as  the  frequency.)  If 
a  simple  form  of  frequency  transformer  could  be  devised, 
by  means  of  which  the  frequency  of  the  supply  circuit 
could  be  gradually  altered,  this  would  afford  a  convenient 
means  of  regulating  the  speed  of  induction  motors,  within 
certain  limits,  depending  upon  the  amount  of  the  leakage 
field  with  the  higher  frequencies  (or  low  inductions),  and 
the  permissible  limiting  value  of  the  induction  in  the 
iron  with  the  lower  frequencies  and  speeds. 

By  changing  the  number  of  stator  poles,  it  is  possible 
to  design  a  motor  to  run  at  two  or  more  different  speeds, 
and  tests  made  on  such  motors  have  shown  the  method 
to  be  practicable;  but  it  involves  some  complication  in 


SPEED   REGULATION  24! 

he  stator  windings,  and  the  use  of  special  switches  with 
connections  so  arranged  that,  by  combining  the  coils  in 
various  ways,  the  number  of  poles  may  be  altered. 
Moreover,  since  an  induction  motor  is  usually  designed, 
in  the  first  instance,  with  as  large  a  number  of  poles  as 
the  diameter  of  the  rotor  will  allow  of,  any  increase  in 
this  number,  for  the  purpose  of  giving  a  reduced  speed, 
will  lead  to  a  corresponding  increase  in  the  magnetic 
leakage,  with  its  attendant  disadvantages. 

The  series-parallel  control  of  two  machines  with  rotors, 
coupled  mechanically  to  a  common  shaft,  offers  a  means 
of  operating  efficiently  at  a  speed  corresponding  to  half 
the  frequency  of  the  supply  circuit.  The  method  is 
analogous  to  the  control  of  two  rigidly  connected  D.C. 
shunt  motors  on  a  constant  potential  supply,  with  the 
armature  windings  connected  in  series.  The  speed  of 
two  similar  motors  so  connected  will  be  half  that  of  any 
one  machine  with  full  supply  pressure  across  the  brushes. 
The  change  from  half  to  full  speed  may  be  made  gradual 
by  connecting  the  armature  windings  in  parallel  through 
suitable  external  resistances  across  the  full  supply  vol- 
tage, the  resistance  being  cut  out  in  sections  to  bring 
the  speed  up  to  the  maximum  limit.  In  the  case  of  the 
induction  motor,  the  secondaries  are  of  the  wound  type, 
with  slip  rings.  The  primary  of  motor  No.  i  is  con- 
nected to  the  supply  circuit,  but  the  primary  of  motor 
No.  2  is  fed  by  currents  from  the  secondary  of  No.  i. 
The  frequency  of  the  currents  in  the  secondary  of  an  in- 
duction motor  is  proportional  to  the  slip,  being  zero  for 
zero  slip,  and  equal  to  the  supply  frequency  when  the 
rotor  is  at  rest.  As  the  speed  of  the  motor  set  increases, 
the  frequency  of  the  supply  to  the  stator  terminals  of 
motor  No.  2  will  decrease,  reaching  a  limit  at  half  the  fre- 
quency of  the  supply  circuit.  Thus  a  stable  condition  of 

16 


242  POLYPHASE  INDUCTION   MOTORS 

running  is  reached  at  a  speed  of  synchronism,  which  is  just 
half  the  speed  obtainable  with  short-circuited  rotors  and 
stators  supplied  with  the  full  line  voltage.  To  effect  the 
change  from  half  to  full  speed,  and  obtain  intermediate 
speeds  if  desired  (not  without  loss  of  energy  during  this 
period),  the  primary  of  motor  No.  2  is  connected  directly 
across  the  supply  mains,  while  adjustable  resistances  are 
inserted  in  series  with  the  rotor  windings.  When  these 
resistances  are  entirely  cut  out,  the  speed  of  the  com- 
bined set  will  be  the  same  as  the  maximum  speed  of 
either  motor  with  secondary  short-circuited.  This  method 
of  speed  regulation  is  generally  referred  to  as  "  tandem  " 
control ;  but  those  familiar  with  the  Latin  tongue,  and 
desirous  of  employing  a  less  equivocal  term,  are  at 
liberty  to  use  the  more  pertinent  word  "  concatena- 
tion." 

Returning  to  method  (i)  as  applied  to  the  speed 
regulation  of  induction  motors,  the  effect  of  introducing 
resistance  in  the  rotor  windings  has  already  been  fully 
discussed  (see  article  76,  p.  205).  It  must,  however,  be 
realised  that  the  reduction  in  speed  is  only  obtained  by 
wasting  the  balance  of  the  power  in  the  resistances, 
exactly  as  in  the  case  of  the  direct-current  motor  with 
regulating  rheostat  in  series  with  the  armature.  A 
numerical  example  will  make  this  clear. 

Example. — Consider  an  induction  motor  of  100  brake 
horse-power,  the  efficiency  of  which  is  -9  at  full  load; 
and  assume  that  it  is  required  to  reduce  the  speed  20  per 
cent.  The  power  supplied  to  the  motor  at  full  load  is 

100  x  746  x  —  =  83  kw.  (approx.) 
and  the  power  absorbed  by  the  resistances  inserted  in 


SPEED   REGULATION  243 

rotor   winding,  for  a  speed  reduction  of  20   per   cent. 
must  be 

83  x  20  =  l6-6  kw. 

100 

In  order  to  calculate  the  ohms  required  per  phase,  it 
is  necessary  to  know  the  value  of  the  full-load  current  — 
i.e.,  the  current  corresponding  to  the  maximum  torque  to 
be  exerted  by  the  rotor.  Let  us  assume  a  star-wound 
rotor  with  300  volts  between  any  two  of  the  three  slip 
rings  on  open  circuit.  Then  if  80  kw.  be  taken  as  the 
power  put  into  the  rotor,  the  current  in  each  of  the  three 
arms  of  the  winding  will  be 

!  =  _.  80,000     =  154  amperes. 


Now,  since  the  total  power  dissipated  in  the  resistance 
is  1  6  -6  kw.,  the  watts  absorbed  by  each  of  the  three 
sections  will  be 

=  5>53o. 


Hence  I2  R  (per  phase)  =  5,530, 

which  gives  us  for  the  resistance  of  each  section  of  the 
resistance 

"2      ohms. 


If,  instead  of  absorbing  the  difference  of  pressure  in  a 
resistance,  it  were  possible  to  insert  a  back  E.M.F.  in 
the  rotor  windings,  from  an  outside  source,  exactly  the 
same  result  would  be  obtained.  Such  a  method  has, 
indeed,  been  suggested  ;  but  since  some  form  of  com- 


244  POLYPHASE   INDUCTION    MOTORS 

mutator  is  required,  involving  the  use  of  a  rotor  wound 
somewhat  in  the  same  manner  as  a  direct-current  motor 
armature,  the  machine  becomes  more  costly,  and  loses  its 
peculiar  advantage  of  mechanical  simplicity  owing  to  the 
absence  of  a  commutator. 

87.  Reversing    Direction    of   Revolution.— The 

direction  in  which  a  polyphase  induction  motor  will  run 
is  determined  by  the  direction  of  rotation  of  the  magnetic 
field.  By  reversing  the  direction  in  which  the  field  re- 
volves, the  direction  of  running  of  the  rotor  will  also  be 
reversed.  In  fact,  a  simple  form  of  throw-over  switch 
in  the  connections  to  the  stator  terminals  is  generally  all 
that  is  necessary. 

If  the  supply  is  three-phase,  it  matters  not  whether  the 
coils  are  star  or  mesh  connected,  but,  provided  the  stator 
is  fed  by  only  three  wires,  the  effect  of  crossing  the  con- 
nections to  any  two  of  the  terminals  will  cause  the 
reversal  of  the  rotating  field,  and,  therefore,  of  the  rotor. 

88.  Recapitulation. — We  have  now  studied  the  in- 
duction motor  by  means  of  vector  diagrams,  and  seen 
how  this  type  of  machine  may  be  considered  as  a  special 
case  of  the  alternate-current  transformer,  the  points  of 
difference  being : 

1 .  That  the  secondary  winding  is  free  to  revolve  in  the 
field  produced  by  the  primary,  and  so  generate  a  second 
E  M.F.  called  the  E.M.F.  of  rotation  which — by  acting 
against  the  induced  E.M.F. — limits  the  flow  of  current 
until  this  is  just  sufficient  to  produce  the  required  torque. 

2.  Owing  to  the  necessary  mechanical  clearance,  or 
air-gap    between    primary    and    secondary    cores,    the 
magnetising    current   is    greater   than   in   the    ordinary 
closed-circuit  transformer ;  moreover,  the  total  primary 
current,  under  light  load  conditions,  is  more  nearly  in 


RECAPITULATION  245 

phase  with  the  true  (or  "  wattless  ")  magnetising  com- 
ponent, because  the  "energy"  component  due  to  the 
hysteresis  and  eddy-current  losses  is  comparatively  small. 

3.  The  magnetic  leakage  is  much  greater  than  in  a 
well-designed  static  transformer.  It  is  the  aim  of  the 
designer  to  keep  this  as  small  as  possible,  because  a 
large  leakage  flux  means  small  overload  capacity, 
although  when  the  machine  is  running  lightly  loaded — 
i.e.,  when  the  rotor  currents  are  small  —  nearly  all  the 
magnetic  flux  passes  through  the  rotor,  and  the  leakage 
is  in  any  case  small. 

This  leads  up  to  the  consideration  of  what  is  the  best 
shape  of  slot  through  which  the  stator  windings  are 
threaded.  In  this  country,  and  on  the  Continent,  com- 
pletely closed  slots  with  an  exceedingly  thin  metal  bridge 
adjoining  the  air-gap  are  not  unusual.  In  America  a 
small  gap  at  the  top  of  the  slot  is  generally  provided, 
the  object  being  to  reduce  the  magnetic  leakage  and 
avoid  the  high  labour  cost  of  winding.  This  has  little 
effect  upon  the  running  qualities  of  the  motor  (except  as 
regards  the  overload  capacity) ;  but  it  admits  of  a  greater 
starting  torque  being  obtained,  or  the  same  starting  torque 
with  a  reduced  primary  current,  owing  to  the  fact  that  a 
larger  proportion  of  the  total  primary  flux  passes  through 
the  rotor.  It  must,  however,  not  be  overlooked  that  open 
slots,  although  they  may  reduce  the  leakage,  generally 
increase  the  reluctance  of  the  magnetic  circuit  through 
the  rotor,  and  consequently  the  amount  of  the  "wattless" 
magnetising  current.  They  also  lead  to  "  pulsation  "  of 
the  flux,  owing  to  want  of  uniformity  in  the  air-gap  flux 
distribution,  an  effect  which  is  practically  absent  when 
closed  slots  are  used,  and  which  is  largely  overcome  by 
using  partially  closed  slots,  as  the  "  fringing  "  with  these 
slots  makes  them  almost  equivalent  to  closed  slots,  in 


246  POLYPHASE   INDUCTION   MOTORS 

that  they  produce  no  appreciable  flux  pulsation  in  the 
teeth.  Thus  higher  flux  densities  may  be  used  in  closed 
slot  machines  than  in  open  slot  machines,  and  where  the 
cost  of  winding  is  not  excessive,  closed  slot  machines  will 
be  the  cheaper  as  well  as  the  lighter  for  the  same  output 
and  speed.  For  the  same  weight  and  size,  the  closed  slot 
machine  will  generally  have  the  better  characteristics. 

In  conclusion,  although  the  writer  prefers  to  consider 
the  polyphase  induction  motor  as  a  modified  transformer 
with  a  closed  secondary  free  to  revolve  in  two  alternating 
magnetic  fields,  differing  in  direction  by  90  "  electrical 
space-degrees,"  and  with  a  phase  difference  of  a  quarter 
period,  it  does  not  follow  that  it  is  incorrect  or  more 
difficult  to  treat  the  subject  from  the  "  rotating  field  " 
point  of  view.  Indeed,  if  the  reader  will  again  turn  to 
article  31,  Chapter  III.,  he  will  probably  find  that  his 
conception  of  a  short-circuited  rotor  being  dragged  round 
by  the  revolving  field  is  clearer  to  him  than  before  read- 
ing the  present  chapter. 


CHAPTER  VIII 

ASYNCHRONOUS  GENERATORS,  FREQUENCY  CONVERTERS, 
COMPENSATED  INDUCTION  MOTORS,  AND  ROTARY 
CONVERTERS 

89.  The  Polyphase  Induction  Motor  used  as  a 
Generator. — It  has  been  explained  in  the  last  chapter 
how,  when  the  load  is  put  on  an  induction  motor,  the 

"  slip,"  or  the  difference  in  speed  between  rotor  and  magnetic 
field  increases,  the  reason  being  that,  with  a  demand  for 
increased  torque,  a  greater  rotor  current  is  required  to 
meet  it,  and — with  an  approximately  constant  speed  and 
strength  of  the  rotating  field — this  increased  current  can 
only  be  obtained  by  a  lowering  of  the  back  E.M.F.  of 
rotation  in  the  rotor  conductors  such  as  will  be  the 
immediate  result  of  increased  "  slip." 

When  the  motor  is  running  light,  the  slip  is  small — 
that  is  to  say,  the  rotor  is  running  nearly,  but  not  quite, 
at  synchronous  speed — this  being  due  to  the  fact  that  a 
small  rotor  current  must  necessarily  flow,  or  sufficient 
torque  will  not  be  produced  to  overcome  bearing  friction, 
windage,  etc. 

Imagine  a  small  auxiliary  motor  coupled  to  the  shaft, 
and  supplied  with  power  from  an  outside  source.     This' 
motor  can  be  arranged  to  increase  the  speed  of  the  rotor 
until  this  is  exactly  equal  to  the  speed  of  synchronism : 
the  rotor  current  will  then  be  zero,  but  the  auxiliary 

247 


248  ASYNCHRONOUS  GENERATORS,    ETC. 

motor  will  be  supplying  the  power  to  overcome  the  light- 
load  losses  referred  to  above. 

Suppose,  now,  that  the  speed  of  the  auxiliary  motor  be 
increased  so  as  to  drive  the  rotor  of  the  polyphase 
machine  at  a  speed  slightly  above  synchronism — i.e.,  greater 
than  that  of  the  rotating  field.  The  result  will  be  that  the 
E.M.F.  generated  in  the  rotor  conductors,  due  to  the 
cutting  of  the  magnetic  lines,  will  now  be  greater  than 
the  induced  E.M.F. ;  and  currents  will,  therefore,  flow 
in  the  rotor  windings  in  a  direction  exactly  opposite  to 
that  obtained  under  ordinary  conditions  of  working.  The 
negative  rotor  current  necessarily  implies  a  negative 
primary  current ;  or,  in  other  words,  the  component  of 
the  total  primary  current  which  balances  the  rotor 
current  will  now  be  forced  to  flow  back  into  the  supply 
circuit,  against  the  primary  impressed  E.M.F. ;  and  the  in- 
duction motor — when  mechanically  driven  at  a  speed 
above  synchronism — becomes  a  generator  of  electric  power. 

Such  machines  are  known  as  asynchronous  generators, 
to  distinguish  them  from  the  more  usual  form  of  syn- 
chronous alternator  or  polyphase  generator.  It  should 
be  particularly  noted  that  the  periodicity  of  the  currents 
obtained  from  the  terminals  of  an  asynchronous  generator 
is  independent  of  the  rotor  speed,  being  determined  solely  by 
the  frequency  of  the  E.M.F.  supply  brought  to  the 
terminals,  for  the  purpose  of  producing  the  magnetic 
field,  and  without  which  these  machines  would  not  work.  / 

90.  Vector  Diagrams  of  Asynchronous  Gener- 
ator.— In  Fig.  91  the  vector  diagram  of  an  induction 
motor  has  been  drawn  to  illustrate  the  effect  of  mechanic- 
ally driving  the  rotor  at  a  speed  slightly  above  syn- 
chronism, and  so  transforming  the  machine  into  a 
generator  of  electric  power.  This  diagram  should  be 
compared  with  Fig.  82  on  p.  209,  and  it  will  be  seen 


DIAGRAMS  OF  ASYNCHRONOUS   GENERATOR      249 

how  the  difference  lies  in  the  rotor  current,  I2,  being 
drawn  exactly  equal  to  the  current  I2  in  Fig.  82,  but  in 
an  opposite  direction.  The  same  assumptions  have  been 
made  in  order  to  simplify  the  diagram,  these  being 
(i)  that  the  resistance  of  stator  windings  is  negligible, 
and  (2)  that  the  rotor  windings  are  without  self-induction. 
This  last  assumption  involves  the  idea  of  the  whole  of  the 


A- 


leakdge  magnetism  enclosing-  the  stator  windings  only,  and 
not  passing  through  the  iron  core  on  which  the  rotor  conductors 
are  wound.  The  current  vector,  Olm,  is,  therefore,  the 
magnetising  component  of  that  portion  of  the  total 
magnetic  flux  which  passes  through  both  rotor  and  stator 
windings.  It  induces  an  E.M.F.,  E2,  in  the  rotor  coils; 
and,  since  these  coils  are  being  revolved  at  a  speed  above 
synchronism,  the  E.M.F.,  E2  Er,  due  to  the  rotation  in  the 


250          ASYNCHRONOUS   GENERATORS,   ETC. 

flux  at  right  angles  to  the  inducing  flux,  is  now  greater 
than  the  induced  E.M.F.,  O  E2,  the  result  being  a  current, 
O  I2,  flowing  against  the  induced  E.M.F.  The  balancing 
component  of  the  stator  current  must,  of  course,  be  O  Ip 
exactly  equal  but  opposite  to  O  I2 :  the  E.M.F.  due  to 
the  leakage  field  will  be  O  E3,  which  must  now  be  drawn 
below  the  line  A  A'  instead  of  above,  as  in  Fig.  82 ;  and, 
by  completing  the  diagram,  we  obtain  O  E  as  the  total 
necessary  impressed  E.M.F.  corresponding  to  the  current, 
O  I2,  flowing  in  the  rotor. 

We  now  see  how  the  product  of  the  vectors  O  E  and 
O  I  is  negative,  since  the  current  in  stator  windings  is 
flowing  against  the  impressed  E.M.F.,  and  therefore 
returning  power  to  the  circuit.  All  the  vectors  are  of 
the  same  length  as  in  Fig.  82,  and  the  angles  6  are  equal. 
It  follows  that  the  power  given  back  to  the  circuit, 
according  to  Fig.  91,  is  the  same  as  that  which  was 
supplied  to  the  motor  according  to  Fig.  82,  and  the 
power  supplying  the  PR  losses  in  the  rotor — represented 
by  the  product  O  I2  x  O  Er — is  taken  from  the  driving 
source,  and  transmitted  to  the  rotor  through  the  shaft. 

In  Fig.  92  the  asynchronous  machine,  A,  and  the  syn- 
chronous machine,  Z,  are  shown  coupled  electrically  on 
a  load  which  may  consist  of  incandescent  lamps  or  induc- 
tion motors,  or  both.  The  connections  are  shown  for 
one  phase  only,  so  as  not  to  complicate  the  diagram. 
The  current  through  the  stator  coils  of  machine  A  is  I, 
corresponding  to  the  vector  O  I  in  Fig.  91.  This  may 
be  considered  as  made  up  of  two  components — the 
current,  L,  going  to  the  load,  and  the  current,  S,  passing 
through  the  armature  windings  of  the  synchronous 
machine,  Z. 

We  shall  suppose  the  condition  of  things  to  be  as 
represented  by  Fig.  91 — that  is  to  say,  the  synchronous 


DIAGRAMS   OF  ASYNCHRONOUS  GENERATOR      251 

machine,  Z,  runs  at  a  constant  speed,  and  produces  a 
definite  pressure,  E,  at  terminals,  while  the  machine  A 
runs  at  a  constant  speed  somewhat  above  the  speed  of 
synchronism,  which  results  in  a  definite  value  of  the 
current  O  I  and  a  definite  phase  angle  0  between  current 
and  impressed  E.M.F. 

In  Fig.  93  the  diagram  Fig.  91  has  been  re-drawn  in 
order  to  show  merely  the  current  O  I  making  an  angle  0 
with  the  pressure  vector,  O  E",  which  may  be  con- 
sidered as  the  E.M.F.  at  the  terminals  of  the  load,  while 
O  E  represents  the  E.M.F.  generated  by  the  synchronous 


Load 


FIG.  92. 

machine.  Drop  a  perpendicular,  I  Llf  on  to  O  E",  and 
note  that,  if  the  load  is  non-inductive  and  equal  to  the  pro- 
duct O  E"  x  O  I  cos  0,  then  the  asynchronous  machine, 
A,  is  taking  the  whole  of  the  load,  while  the  synchronous 
machine,  Z,  acts  merely  the  part  of  an  exciter,  and  does 
no  work,  since  the  current  component,  O  Sr  is  "wattless." 
If  the  load  is  greater  than  this  amount,  and  equal  to 
O  E"  x  O  L2,  then  machine  A  will  still  do  its  share, 
and  give  the  whole  of  its  output  to  the  load;  but  the 
balance  required  (which  is  equal  to  O  E  x  L2  Lj)  must 


252 


ASYNCHRONOUS  GENERATORS,    ETC. 


be  supplied  by  the  machine  Z,  the  total  output  of  which 
may  be  written  (O  E)  x  (L2  I)  cos  /?,  where  L2  I  is  the 
current  S  in  the  diagram  Fig.  92. 

If  the  external  load  is  less  than  the  full  output  of  the 
asynchronous  machine,  as,  for  instance,  O  E"  x  O  L3, 
then  the  current,  S,  in  the  armature  of  the  synchronous 
machine  is  represented  by  the  vector  L3  I,  which  makes 
an  angle  with  O  E  greater  than  90  degrees.  It  follows 
that  the  power  given  out  by  machine  Z  is  negative — i.e., 


0| 


L3    L!     L2 


FIG.  93. 

this  machine  is  being  driven  as  a  motor  by  the  machine  A, 
the  power  expended  in  doing  this  being  equal  to  O  E"  x 
L»  Lx. 

Let  us  now  see  what  is  the  effect  of  an  inductive  load. 
The  diagram  Fig.  94  has  been  drawn  in  a  similar  manner 
to  Fig.  93,  but  the  load  current  now  lags  behind  the 
pressure  O  E",  by  a  certain  angle  </>.  If  the  load  current 
is  O  Lj,  such  that  I  Lx  is  at  right  angles  to  O  E",  then 
the  power  absorbed  is  the  same  as  before,  being  equal  to 
O  E"  x  O  P  or  to  O  E"  x  O  I  cos  0,  which  is  the  total 


DIAGRAMS  OF  ASYNCHRONOUS  GENERATOR      253 

output  of  the  asynchronous  machine — for  the  particular 
conditions  as  regards  speed  and  excitation  that  we  have 
assumed.  But  the  "  wattless  "  current,  O  S15  from  the 
armature  of  the  synchronous  machine  is  now  greater  than 
O  Sx  in  Fig.  93  by  an  amount  P  Lx,  which  is  exactly 
equal  to  the  wattless  component  of  the  load  current ; 
from  which  it  is  clear  that  any  "  wattless "  currents 
required,  owing  to  the  load  having  self-induction,  cannot 
be  provided  by  the  asynchronous  machine,  but  must  be 


FIG.  94. 

supplied  by  the  synchronous  generator  in  addition  to  the 
wattless  exciting  current  required  by  the  former.  If  O  L2 
is  the  load  current,  L2  I  is  the  current  S  (Fig.  92)  from 
machine  Z  ;  while,  if  the  load  current  is  O  L3,  the  power 
absorbed  is  less  than  the  full  output  of  the  machine  A, 
and  the  synchronous  machine  is  again  being  driven  as 
a  motor. 

91.  Conclusions  regarding  the  Induction  Motor 
used  as  a  Generator.-^Since  the  frequency  is  inde- 


254          ASYNCHRONOUS  GENERATORS,   ETC. 

pendent  of  the  speed  of  an  asynchronous  generator  (being 
determined  solely  by  the  speed  of  the  synchronous 
machine  which  supplies  the  magnetising  currents  to 
the  stator  winding),  it  follows  that  machines  of  this 
class  can  be  paralleled  without  the  necessity  of  regu- 
lating the  speed  to  such  a  nicety  as  when  paralleling 
the  more  common  type  of  alternating-current  generators. 
In  fact,  the  various  units  have  merely  to  be  run  up 
to  approximately  synchronous  speed,  and  switched  on 
to  the  'bus  bars,  much  in  the  same  manner  as  when 
coupling  direct-current  dynamos.  The  proportion  of  the 
total  load  taken  up  by  each  machine  (where  several  are 
coupled  in  parallel)  will  depend  solely  upon  the  relative 
speeds  at  which  the  various  units  are  driven. 

It  would  almost  seem  as  if  this  convenience  in  paral- 
leling were  the  only  recommendation  for  machines  of 
this  class. 

They  have  the  disadvantage  of  requiring  a  synchronous 
generator,  always  running,  to  fix  the  frequency  and  pro- 
vide the  wattless  magnetising  currents ;  moreover,  since 
the  asynchronous  machines  are  capable  of  supplying  the 
external  circuit  only  with  energy  currents,  the  synchronous 
generator  may  have  to  be  of  a  very  large  size  if  the  load 
is  inductive,  seeing  that  it  will  be  called  upon  to  supply 
the  wattless  component  of  the  total  load  current,  in 
addition  to  the  magnetising  currents  of  the  other  gener- 
v  ators. 

With  the  advent  of  the  compensated  induction  motor 
as  introduced  and  developed  by  Mr.  Alex.  Heyland  and 
others,  there  is  a  possibility  of  such  machines — when 
used  as  generators — proving  a  commercial  success,  owing 
mainly  to  the  fact  that  they  can  be  loaded  inductively, 
thus  bringing  the  size  of  the  synchronous  exciter  within 
reasonable  limits.  It  is  not  proposed  to  discuss  this 


INDUCTION    MOTOR   USED  AS   A  GENERATOR      255 

question  at  greater  length;  but  since  there  is  a  possi- 
bility—even if  little  probability — of  the  compensated 
polyphase  induction  motor  with  commutator  being  used 
extensively  in  the  near  future,  the  theory  and  properties 
of  such  motors  will  be  briefly  dealt  with  in  article  93. 

There  is  one  point  in  connection  with  the  asynchronous 
generator  which  should,  perhaps,  be  mentioned,  and  this 
is  the  peculiar  effect  of  a  heavy  overload.  If  the  speed 
of  the  prime  mover  be  increased  beyond  a  certain  limit, 
the  proportion  of  the  load  taken  by  the  generator  will 
begin  to  fall  off,  and  unless  suitable  precautions  are  taken, 
e  is  a  possibility  of  the  engine  "  running  away." 
This  will  be  evident  from  a  study  of  Fig.  91,  which, 
as  already  pointed  out,  differs  but  little  from  the  corre- 
sponding diagram  of  the  same  machine  used  as  a  motor. 
In  the  case  of  the  motor  it  has  been  shown  how,  when 
the  torque  exceeds  a  certain  limit,  the  leakage  field 
becomes  so  great  that  the  machine  is  unable  to  respond 
to  the  additional  call  upon  it,  and  falls  out  of  step.  In 
the  generator  the  effect  is  similar,  and  the  overload 
capacity  of  the  machine  is  reached  when  the  speed 
exceeds  the  speed  of  synchronism  by  an  amount  approxi- 
mately equal  to  the  "  slip  "  corresponding  to  the  maxi- 
mum output  of  the  same  machine  when  run  as  a  motor. 

92.  Frequency  Converters. — Although  the  frequency 
of  the  currents  in  the  primary  circuit  of  an  induction 
motor  remains  unaltered  whatever  may  be  the  speed  of 
the  rotor,  the  frequency  of  the  currents  in  the  secondary 
windings  is  dependent  upon  the  speed  of  rotation.  Thus, 
when  the  rotor  is  at  rest,  the  machine  may  be  looked 
upon  as  a  static  transformer  of  inferior  design,  with 
rather  poor  regulation  because  of  the  increased  leakage 
due  to  the  air-gap  and  necessary  separation  between 
primary  and  secondary  windings ;  the  periodicity  of  the 


256          ASYNCHRONOUS   GENERATORS,    ETC. 

secondary  E.M.F.  will,  however,  be  the  same  as  that  of 
the  primary  impressed  E.M.F.  If,  now,  a  certain  speed 
of  rotation  be  imparted  to  the  rotor  by  means  of  an  inde- 
pendent motor  coupled  to  the  shaft,  the  E.M.F.  in  the 
secondary  windings  will  depend  upon  the  relative  speed 
of  rotor  and  revolving  field.  In  this  connection,  a  mental 
picture  of  the  revolving  field  is  perhaps  more  useful  than 
the  recently-discussed  equivalent  of  two  flux  components 
with  an  angular  displacement  of  90  electrical  (space) 
degrees.  If  the  rotor  is  provided  with  slip-rings,  the 
frequency  of  the  current  taken  from  the  collecting 
brushes  v/ill  be  directly  proportional  to  the  difference 
in  speed  between  revolving  field  and  rotor.  If  the  rotor 
is  driven  backward — i.e.,  in  a  direction  contrary  to  that 
of  the  rotating  field — the  secondary  frequency  will  be 
higher  than  that  of  the  primary  circuit. 

In  order  to  obtain  a  constant  frequency  at  secondary 
terminals,  the  slip  (whether  positive  or  negative)  must 
bear  a  definite  relation  to  the  primary  frequency,  and 
the  motor  driving  the  shaft  must  therefore  be  a  constant 
speed  machine,  such  as  a  synchronous  motor  connected 
to  the  primary  supply  circuit.  The  ratio  of  frequencies 
is  obviously 

secondary  frequency       "  slip  "  revolutions 
primary  frequency     "  synchronous  speed 

In  regard  to  the  power  capacity  of  the  auxiliary  motor, 
when  the  slip  is  100  per  cent. — i.e.,  when  the  rotor  is  at  a 
standstill — the  auxiliary  motor  is  doing  no  work  ;  and 
when  the  slip  is  200  per  cent. — i.e.,  when  the  rotor  is 
driven  backward  at  a  speed  corresponding  to  that  of  the 
revolving  field — the  auxiliary  motor  is  providing  the  same 
power  as  the  primary  circuit  supplies  through  trans- 
former action.  This  should  be  evident  without  proof; 


"COMPENSATED"  POLYPHASE  MOTORS     257 

but,  bearing  in  mind  that  the  total  output  plus  the  losses 
must  exactly  equal  the  joint  capacity  of  the  auxiliary 
driving  motor  and  the  converter  proper,  and  further 
that  torque  x  slip  represents  the  power  demanded  of 
the  auxiliary  motor,  it  will  be  seen  that  the  ratio  of 
primary  to  secondary  frequency  is  also  the  ratio 
of  "  transformer  power  "  to  total  power,  the  difference 
between  these  quantities  being  provided  by  the  auxiliary 
driving  motor. 

As  a  frequency  changer  per  se,  the  induction  motor 
with  independently  driven  rotor,  as  above  described, 
is  rarely  met  with  in  connection  with  practical  under- 
takings ;  but  when  combined  with  the  rotary  converter, 
it  forms  the  input  unit  of  the  motor  converter,  a  type  of 
machine  that  will  be  referred  to  in  article  100. 

93.  "Compensated"    Polyphase    Motors    with 

Commutators. — To  Mr.  Alex.  Hey  land  must  be  given 
credit  for  the  idea  of  supplying  the  magnetising  currents 
of  an  induction  motor  through  the  rotor  windings  instead 
of  through  the  stator  coils.  This  ingenious  method  has 
the  effect  of  improving  the  power  factor,  which  can  thus 
be  brought  up  to  about  100  per  cent,  for  all  loads. 
Other  workers,  including  Messrs.  Gorges,  Latour,  and 
Osnos,  have  devised  machines  somewhat  on  the  same 
lines;  but  since  there  is  no  difference  in  the  principle 
involved,  we  shall  content  ourselves  with  briefly  con- 
sidering how,  by  feeding  the  magnetising  currents 
through  a  commutator  into  the  rotor  windings,  it  is 
possible  to  compensate  for  phase  displacement,  and 
bring  the  power  factor,  even  at  light  loads,  approxi- 
mately up  to  unity. 

In  Fig.  95,  the  rotor  is  supposed  to  be  of  the  wound 
type,  and  provided  with  a  commutator  in  all  respects 
similar  to  that  of  a  direct-current  bipolar  machine. 


2$8  ASYNCHRONOUS   GENERATORS,   ETC. 

Imagine  the  rotating  field  to  be  produced  by  two  alter- 
nating fields,  one  acting  in  the  direction  B  B',  and  the 
other — differing  in  phase  by  a  quarter  period — acting  in 
the  direction  A  A',  at  right  angles  to  B  B' ;  this  being 
the  method  of  treating  the  subject  which  was  justified 


and  explained  in  the  last  chapter,  when  describing  th 
action  of  the  ordinary  induction  motor. 

Consider,  first,  what  goes  on  in  the  phase  B  B'  only. 
The  stator  windings,  in  this  phase,  act  merely  as  the 
primary  of  a  static  transformer ;  and,  if  we  suppose  the 
rotor  or  secondary  windings  to  be  without  any  short- 


"COMPENSATED"    POLYPHASE   MOTORS        259 

circuiting  connections,  the  current  that  will  flow  in  the 
stator  windings  when  these  are  connected  to  the  supply 
mains  will  be  the  magnetising  current ;  and  although  the 
true  magnetising  watts — i.e.,  the  product  of  the  current 
and  resultant  volts — may  be  small,  the  apparent  watts  are 
relatively  great,  since  they  are  equal  to  the  product  of  the 
current  and  the  full  pressure  of  the  supply. 

Suppose  now  that,  with  the  armature  at  rest,  and  with 
the  stator  windings  disconnected  from  the  supply,  an 
alternating  current  of  the  same  frequency,  but  suitably 
reduced  pressure,  is  fed  into  the  brushes,  m  and  n,  bearing 
on  the  commutator  at  the  two  ends  of  a  diameter  lying 
on  the  magnetic  axis,  B  B',  as  shown  in  Fig  95.  Th 
current  will  divide  itself  between  the  two  halves  of  the 
winding,  and  produce  an  alternating  field  in  the  direction 
B  B',  which — if  the  pressure  be  adjusted  to  suit  the 
number  of  turns  in  the  rotor  winding — may  be  made  of 
the  same  strength  as  the  field  originally  produced  by  the 
stator  winding.  But  with  the  rotor  at  vest  the  power  factor 
has  not  been  improved,  since  the  rotor  coils  may  now  be 
considered  as  constituting  the  primary  winding  of  a 
transformer,  and  the  same  trouble  occurs  —  i.e.,  the 
magnetising  current  produces  the  magnetic  flux,  which, 
in  its  turn,  gives  rise  to  a  back  E.M.F.  of  self-induction 
very  much  in  excess  of  the  resultant  E.M.F.  required  to 
overcome  the  resistance  of  the  windings,  and  this  neces- 
sitates an  impressed  E.M.F.  not  only  slightly  greater  than 
the  induced  E.M.F.,  but — on  account  of  the  relatively 
high  reluctance  of  the  air  gaps — nearly  90  degrees  in 
advance  of  the  current,  as  regards  phase. 

Now  imagine  the  rotor  to  be  revolving  at  synchronous 
speed.  The  brushes  bear  on  the  commutator,  but  retain 
a  fixed  position  in  space,  the  result  being  that  the 
alternating  current  fed  into  the  brushes  will  produce  an 


260  ASYNCHRONOUS   GENERATORS,    ETC. 

alternating  flux  in  the  same  direction,  and  of  the  same 
frequency  as  before ;  this  frequency  being  quite  independent 
of  the  speed  at  which  the  rotor  is  revolving,  and  depending 
merely  upon  the  frequency  of  the  supply  connected  across 
the  brushes.  It  is  true  that  the  frequency  of  the  current 
in  any  particular  portion  of  the  rotor  winding  will  depend 
upon  the  speed  of  revolution — and,  in  fact,  for  the  con- 
dition of  synchronous  speed  that  we  are  considering,  the 
current  in  the  rotor  conductors  would  be  a  continuous 
one;  but  the  point  which  concerns  us  at  present  is  that, 
as  the  rotor  revolves,  the  conductors  lying  between  the 
brushes,  m  and  n,  are  cutting  the  field  A  A'  at  such  a 
rate  as  to  produce  an  alternating  E.M.F.  of  rotation 
exactly  equal,  but  opposite  in  direction,  to  the  alternating 
E.M.F.  of  induction  due  to  the  field  B  B'.  No  further 
difficulty  need  therefore  be  experienced  in  getting  the 
magnetising  current  through  the  rotor  windings.  The 
E.M.F.  across  the  brushes,  m  and  n,  no  longer  requires 
to  be  of  an  abnormally  great  value,  since  the  induced 
back  E.M.F.  is  now  balanced  by  the  E.M.F.  of  rotation. 
A  few  volts  only  are  needed  to  overcome  the  ohmic 
resistance  of  the  windings,  and  provide  the  necessary 
magnetising  current.  Moreover,  this  pressure  will  now 
be  in  phase  with  the  magnetising  current,  and  not  nearly 
90  degrees  in  advance,  as  was  necessary  so  long  as  the 
back  E.M.F.  of  self-induction  was  a  factor  to  be  reckoned 
with.  It  follows  that  the  magnetising  watts  are  now  true 
watts,  and,  by  regulating  the  pressure  across  the  brushes, 
m  and  n,  until  the  magnetising  ampere  turns  are  equal  to 
those  which  otherwise  would  have  to  be  provided  by  the 
stator  current,  the  latter  may  be  reduced  to  nearly  zero, 
thus  bringing  the  power  factor  (even  with  the  machine 
running  light)  to  a  value  very  near  to  unity. 

When  the  motor  is  loaded,  the  magnetising  current 


"COMPENSATED"    POLYPHASE   MOTORS        26l 

requires  to  be  of  approximately  the  same  value  as  on 
open  circuit,  and  the  power  factor  can  be  maintained  at 
nearly  100  per  cent,  for  all  loads.  It  is  evident  that,  as 
the  load  comes  on  the  motor,  the  speed  is  no  longer  that 
of  synchronism;  but  it  does  not  differ  much  from  the 
speed  of  the  rotating  field,  and  at  full  load,  with  a  "  slip  " 
of,  say,  3  per  cent,  the  currents  in  the  rotor  conductors, 
although  no  longer  direct  currents,  are  of  such  a  low 
periodicity  that  self-induction  troubles  do  not  arise. 

This  is  the  theory  of  the  compensated  induction  motor ; 
it  will  be  understood  that  the  exact  pressure  across  the 
brushes  should  be  adjusted  experimentally  for  each 
particular  machine,  to  suit  the  supply  voltage.  If  the 
pressure  is -too  low,  the  exciting  current  in  the  rotor 
windings  will  be  insufficient  to  provide  the  full  amount  of 
the  flux,  and  it  will,  therefore,  be  supplemented  by  a 
small  magnetising  component  of  the  stator  current,  which 
means  that  the  power  factor  will  still  be  less  than  unity. 
If  too  great  a  current  is  put  into  the  rotor  windings,  this 
will  give  rise  to  a  leading  component  of  the  primary 
current  of  such  a  value  as  to  neutralise  the  excess  of 
magnetising  current  in  the  rotor,  and  maintain  the 
resultant  magnetising  ampere  turns  at  the  exact  value 
required  to  produce  the  necessary  back  E.M.F.  in  the 
stator  windings. 

Although  it  is  not  proposed  to  enter  into  details  of 
construction,  or  to  discuss  the  various  devices  by  means 
of  which  this  theory  of  compensation  may  be  applied  in 
practice,  this  article  would  hardly  be  complete  without  a 
reference  to  the  actual  method  of  compensating  induction 
motors  originally  devised  by  Mr.  Heyland. 

In  Fig.  96,  the  stator  windings  of  the  motor  are  shown 
star-connected  off  the  supply  mains.  The  three  rotor 
coils,  C  C  C,  are  also  star-connected,  being  short-circuited 


262  ASYNCHRONOUS   GENERATORS,   ETC. 

at  their  outer  ends  by  the  heavy  connection,  S,  while  their 
inner  ends  are  joined  through  the  low  resistance,  R.  This 
resistance  actually  takes  the  form  of  small  V-shaped 


Supply  Mains 


FIG.  96. 

connectors  of  low-resistance  material  joining  contiguous 
segments  of  the  commutator ;  and  the  currents  that  are 
brought  to  the  brushes,  B  B  B,  will  circulate  in  the  rotor 


"  COMPENSATED  "    POLYPHASE   MOTORS        263 

coils,  C,  in  such  a  direction  as  to  produce  a  rotary 
magnetic  field  synchronous  with  the  fielcf  that  would 
otherwise  be  produced  by  the  magnetising  components 
of  the  stator  currents.  The  magnetising  currents  led  into 
the  brushes  are  obtained  from  the  secondary  of  a  small 
transformer,  T,  of  which  the  primary  is  connected  to  the 
same  source  of  supply  as  the  stator  windings.  It  will  be 
seen  that  the  primary  coils  of  this  transformer  are  star- 
connected,  while  the  secondary  coils  are  mesh-connected  ; 
the  object  of  this  being  to  provide  currents  to  the  brushes, 
B,  which,  instead  of  being  in  phase  with  the  supply 
pressure,  are  go  degrees  out  of  phase,  and,  therefore,  of 
the  same  phase  as  the  required  magnetic  flux. 

It  is  not  necessary  to  use  the  same  winding  for  the 
magnetising  currents  as  for  the  circulation  of  the  currents 
which  arise  when  the  load  is  put  on  the  machine,  and  in 
the  Heyland  type  of  motor  of  later  design  a  double  three- 
phase  winding  is  used  on  the  rotor  with  advantage. 

Such  machines  when  used  as  asynchronous  generators 
have  undoubted  advantages  over  the  ordinary  type  of 
machine,  and  it  will  be  readily  understood  that  there  are 
no  difficulties  in  the  way  of  compounding  these  com- 
pensated generators,  since  the  necessary  currents  for 
strengthening  the  field  can  be  taken  off  suitable  current 
transformers,  and  led  into  the  commutator  through 
brushes  in  the  same  manner  as  the  compensating 
currents. 

Although  the  compensated  and  compounded  polyphase 
induction  motor  is  undoubtedly  interesting  and  instruc- 
tive from  the  scientific  point  of  view,  it  is  doubtful 
whether  it  has  come  to  stay,  because  unless  the  advan- 
tage of  an  improved  power  factor  is  worth  the  increased 
complication  and  expense  of  manufacture,  the  machine 
gan  never  prove  a  commercial  success. 


264  ASYNCHRONOUS   GENERATORS,    ETC. 

It  must  not  be  overlooked  that  the  addition  of  a  com- 
mutator, witfi  all  the  increased  difficulties  of  winding,  is 
out  of  the  question  for  small  machines ;  and  when  we 
have  to  deal  with  large  machines,  the  power  factor  of 
these  at  full  load  is  usually  over  90  per  cent.,  and  in  order 
to  raise  this  to  100  per  cent,  it  is  a  doubtful  point  whether 
the  increased  cost  and  increased  liability  to  break  down 
are  justified. 

94.  Synchronous  Rotary  Converters. — The  syn- 
chronous converter  is  a  machine  provided  with  a  com- 
mutator, which  receives  alternating  currents  and  delivers 
continuous  currents  all  through  the  one  set  of  armature 
coils.  It  is  generally  used  for  transforming  polyphase 
into  direct  currents,  the  polyphase  currents  being  led 
into  the  armature  windings  through  slip  rings,  while  the 
direct  current  is  collected  by  brushes  bearing  on  the  com- 
mutator. The  machine  may,  therefore,  be  considered 
as  a  combined  synchronous  motor  and  direct-current 
generator. 

When  treating  of  polyphase  generators,  it  was  shown 
how  the  sides  of  a  closed  polygon  of  vectors  represent 
the  respective  alternating  E.M.F.s  in  the  various  (mesh- 
connected)  armature  coils.  Thus,  in  Fig.  97  (a),  the 
vectors  A  B,  B  C,  and  C  A,  forming  the  sides  of  an 
equilateral  triangle,  represent,  by  their  magnitude  and 
direction,  the  amount  and  phase  relations  of  the  E.M.F.s 
in  the  three  windings  of  a  delta-connected  three-phase 
generator  or  motor. 

In  Fig.  97  (b)  and  97  (c),  the  vector  diagrams  have 
been  drawn  for  four-  and  five-phase  machines  respec- 
tively, while  the  dotted  hexagon  in  Fig.  97  (a)  refers  to  a 
six-phase  machine.  The  fact  that  all  these  figures  are 
closed  polygons  is  an  indication  that  the  sum  of  all 
E.M.F.s  acting  in  the  armature  winding  is  zero,  and 


SYNCHRONOUS   ROTARY   CONVERTERS         265 

consequently  currents,  such  as  may  properly  be  repre- 
sented by  vector  diagrams,  will  not  circulate  in  the  coils 
when  the  external  circuit  is  open. 

Now,  since  A  B,  B  C,  etc.,  represent  the  alternating 


(b) 


FIG.  97. 

E.M.F.s  in  the  various  sections  of  the  winding,  which 
are  all  connected  in  series,  the  pressure  between  any  two 
terminals  of  the  polyphase  circuit  (which  is  equal  to  the 
sum  of  the  component  E.M.F.s)  will  be  indicated  on 
the  vector  diagrams,  both  in  magnitude  and  phase,  by  the 


266  ASYNCHRONOUS   GENERATORS,   ETC. 

straight  line  joining  the  beginning  of  the  first  and  the  end 
of  the  last  vector  of  which  the  sum  is  taken.  Thus, 
in  Fig.  97  (£),  the  pressure  between  opposite  terminals 
is  indicated  by  the  lines  A  C  and  B  D,  the  length  of 
which  is  a  measure  of  the  E.M.F.  obtained  by  connecting 
two  of  the  phases  in  series :  the  phase  of  the  resultant 
E.M.F.  is  also  indicated  by  the  angular  position  of  these 
lines  relatively  to  the  component  vectors ;  but  the  direc- 
tion—?.^., whether,  for  instance,  from  A  to  C  or  C  to  A— 
cannot  be  shown  by  an  arrow  on  the  diagram,  as  it  will 
depend  upon  what  is  to  be  considered  as  the  positive 
direction  in  the  external  circuit  joining  the  two 
terminals. 

In  Fig.  97  (<;),  the  pressure  between  the  terminals 
A  and  D  will  be  represented  by  the  length  of  the  line  A  D, 
which  may  be  looked  upon  as  the  sum  of  the  three  vectors 
A  B,  B  C,  and  C  D,  or  of  the  two  vectors  D  E  and  E  A, 
for  these  must  necessarily  balance,  and  give  the  same 
resultant. 

Let  us  now  carry  this  construction  further,  by  increas- 
ing the  number  of  the  sides  of  the  polygon,  and  therefore 
the  number  of  phases,  indefinitely.  This  leads  us  to  the 
diagram  Fig.  97  (d),  which  represents  the  E.M.F.  in  the 
armature  winding  of  the  machine  as  being  due  to  the 
rotation,  in  the  field,  of  an  infinite  number  of  armature 
sections,  in  each  one  of  which  an  alternating  E.M.F.  is 
generated.  The  circle  may  be  considered  as  the  limit  of 
the  polygon  of  vectors  as  the  sides  of  the  polygon  are 
increased  in  number  without  limit ;  and  the  length  of  the 
line  A  B  represents  the  alternating  pressure  obtained 
between  two  tappings  from  the  winding  separated  by 
a  distance  equivalent  to  the  phase  angle  a.  With  regard 
to  the  diameter  A  C,  this  represents  the  pressure  obtained 
between  two  diametrically  opposed  points  of  the  (two-pole) 


SYNCHRONOUS   ROTARY   CONVERTERS          267 

armature  winding;  and  if  we  suppose  the  E.M.F.  wave 
to  follow  the  simple  harmonic  law  of  variation,  the  direct- 
current  pressure  between  the  brushes  bearing  on  the 
commutator  of  a  rotary  converter  or  double-current 
generator  would  be  \/2  times  the  value  given  by  the 
length  AC.* 

The  diagrams,  as  drawn  in  Fig.  97,  are  not  limited  in 
their  application  to  two-pole  machines  ;  it  is  merely 
necessary  to  bear  in  mind  that  the  angles  in  the  diagrams 
are  phase  angles,  and  the  vector  A  C  in  Fig.  97  (b)  or  (d) 
indicates  the  pressure  between  two  points  on  the  actual 
winding  separated  by  a  distance  equal  to  the  pitch  of  the 
poles  :  thus,  it  is  only  in  the  case  of  a  two-pole  machine 
that  these  tappings  would  actually  be  at  the  two  ends  of 
a  diameter. 

A  rotary  converter  can  evidently  not  be  wound  for  any 
desired  pressure  transformation  ;  the  ratio  of  alternating 
to  direct  current  pressure  is  fixed,  being  determined 
solely  by  the  number  of  slip  rings  or  phases.  From  an 
inspection  of  the  diagrams  in  Fig.  97,  it  will  be  seen  that, 
since  the  length  of  the  chord  joining  two  points  on  the 
circle  is  a  measure  of  the  alternating  E.M.F.  between 

*  If  the  E.M.F.  vectors  such  as  A  B  and  A  C  stand  for  the 
maximum  values  of  the  alternating  pressures  instead  of  the 


square  values,  then  the  diameter  of  the  circle  would 
correctly  indicate  the  direct-current  voltage  between  brushes,  no 
multiplier  being  necessary.  It  is  evident  that,  when  tappings  are 
taken  from  diametrically  opposed  points  on  the  winding,  these 
points  must  simultaneously  pass  under  the  direct-current  collecting 
brushes  twice  during  each  revolution  of  the  armature,  and  at  these 
moments  the  instantaneous  value  of  the  alternating  E.M.F.  must 
necessarily  correspond  with  the  direct-current  voltage  :  moreover, 
since  the  brushes  lie  on  the  neutral  axis,  where  the  E.M.F.  is  at  its 
maximum,  it  follows  that  it  is  the  maximum  value  of  the  alternating 
E.M.F.  which  agrees  with  the  direct-current  voltage. 


268 


ASYNCHRONOUS    GENERATORS,   ETC. 


these  points,  the  maximum  value  of  this  E.M.F.  is  given 
by  the  expression 


Emax.  =  2  R  sin     , 


where  R  is  the  radius  of  the  circle  and  a  is  the  phase 
angle  between  the  two  tappings  off  the  armature  winding. 
Expressed  in  terms  of  the  direct-current  voltage,  we 
have 

Emax.  =  D  sin  ^  , 

where  D  —  the  diameter  of  the  circle  —  stands  for  the 
direct-current  pressure  across  brushes.  If  we  assume 
the  sine  curve  wave  form,  the  >/  mean  square  value  of  the 
alternating  pressure  will  be  equal  to  the  above  value 
divided  by  ^  2. 

The  following  table  gives  the  pressure  per  phase  for 
various  numbers  of  slip  rings,  the  direct-current  pressure 
being  taken  as  unity  : 


Number  of  Phases  or 
Slip  Rings. 

Single  phase  or  two  phase 

Sin  f 

E.M.F.  between 
Slip  Rings  = 

i 

•7071 

Three  phase            

J^3-  =  -866 

2 

•6124 

Four  phase 

—  =,  =  7071 

•5000 

Five  phase  ... 
Six  phase     ... 
Twelve  phase 

•5878 
1  =   -5000 
•2588 

•4*57 
•1830 

These  calculated  values  agree  fairly  well  with  figures 
obtained  from  actual  machines. 


THEORY  OF   THE   ROTARY   CONVERTER        269 

95.  Elementary  Theory  of  the  Rotary  Converter. 

— Consider  a  rotary  converter  running  at  full  speed  with 
the  direct-current  circuit  open — i.e.,  without  any  direct 
current  being  taken  out  of  the  armature.  The  condition 
is  merely  that  of  a  synchronous  alternating-current  or 
polyphase  motor  running  light.  Let  us  suppose  the  field 
coils  to  be  excited  from  an  independent  direct-current 
source,  and  note  that — when  the  armature  is  revolving 
at  synchronous  speed — there  is  a  definite  strength  of  field 
required  to  produce  the  necessary  back  E.M.F.  of  rota- 
tion in  the  armature  conductors.  If  the  current  in  the 
field  coils  is  adjusted  to  give  this  particular  strength  of 
field,  the  alternating  currents  fed  into  the  armature 
through  the  slip  rings  from  the  supply  mains  will  be 
very  small,  being  merely  such  as  to  produce  the  torque 
required  to  overcome  the  light  load  losses  ;  moreover, 
this  current  will  be  in  phase  with  the  applied  E.M,F. 

If  we  now  either  increase  or  decrease  the  field  current, 
the  result  will  be  an  alteration  in  the  armature  current, 
which  must  adjust  itself  in  order  that  the  resultant  mag- 
netising ampere  turns  may  be  the  same  as  before  :  thus, 
if  the  current  in  the  field  coils  be  strengthened,  the 
armature  current  will  lead  the  applied  E.M.F.,  while  if 
the  excitation  is  reduced,  the  current  in  the  armature 
will  be  a  lagging  one.* 

Once  the  idea  of  a  constant  resultant  field  excitation  is 
clear,  it  is  an  easy  matter  to  follow  the  actions  which 
take  place  when  a  direct  current  is  taken  out  of  the 

*  Incidentally,  attention  may  be  called  to  the  possibility  of 
regulating  the  power  factor  of  a  polyphase  system  by  over  or  under 
exciting  the  field  coils  of  rotary  converters  connected  to  the  mains. 
These  machines  will,  in  this  respect,  act  in  a  similar  manner  to 
synchronous  motors,  and  they  may  be  made  to  produce  the  effect 
of  either  condensers  or  choking  coils  as  the  case  may  require. 


270  ASYNCHRONOUS   GENERATORS,    ETC. 

armature  through  the  brushes  bearing  on  the  commutator. 
Whatever  portion  of  the  direct  current  passes  through 
sections  of  the  armature  winding  must  be  balanced  at 
every  instant  by  currents  taken  from  the  polyphase  sup- 
ply mains,  in  order  that  the  resultant  field  excitation 
shall  not  be  altered.  It  is  true  that  the  direct  current 
taken  from  the  armature  tends  mainly,  if  not  wholly,  to 
produce  cross  magnetising  ampere  turns  ;  but  these  cross 
magnetising  turns  must  nevertheless  be  balanced  by  the 
polyphase  current  entering  the  armature  through  the 
slip  rings,  because — even  on  the  assumption  of  an  effi- 
ciency of  100  per  cent. — whatever  power  is  taken  out  on 
the  direct-current  side  must  evidently  be  balanced  by  an 
equal  power  supplied  to  the  armature  from  the  polyphase 
side. 

On  account  of  this  balancing  of  the  magnetising  effects 
due  to  the  continuous  and  polyphase  currents,  there  is 
practically  no  armature  reaction  in  the  polyphase  rotary 
converter.  If  the  losses  are  negligible,  and  the  field  cur- 
rent is  adjusted  to  give  the  maximum  power  factor,  there 
is  theoretically  very  little  or  no  armature  reaction,  and 
this  accounts  for  the  fact  that  it  is  not  necessary  to  alter 
the  position  of  the  brushes  with  varying  load.  In  this 
respect  the  rotary  converter  differs  from  the  double- 
current  generator,  because,  in  the  latter,  both  direct  and 
alternating  currents  combine  to  produce  field  distortion.* 

*  The  double-current  generator  is  a  machine  from  the  armature 
of  which  both  alternating  and  continuous  currents  may  be  taken. 
It  has  not  been  considered  in  previous  chapters,  because  it  is  rarely 
used.  It  is  neither  so  satisfactory  nor  so  useful  a  machine  as  the 
synchronous  converter ;  but,  so  far  as  the  component  parts  and 
general  appearance  are  concerned,  it  may  be  looked  upon  as 
a  synchronous  converter  with  the  addition  of  a  shaft  coupling 
or  pulley  through  which  it  receives  mechanical  power  from  an 
outside  source. 


THEORY   OF   THE   ROTARY  CONVERTER        2/1 

It  is  well  known  that,  in  the  continuous-current  trans- 
former with  two  windings  and  two  commutators,  there 
is  no  trouble  with  sparking  at  the  brushes,  owing  to 
the  fact  that  the  cross  magnetising  ampere  turns  of  the 
secondary  winding  are  balanced  by  the  ampere  turns  of 
the  primary  winding ;  and  although,  in  the  polyphase 
rotary  converter,  there  is  only  one  winding,  and  the  current 
taken  out  of  the  machine  is  not  of  the  same  kind  as  the 
current  put  in,  yet,  at  every  instant,  there  is  a  balancing 
action  similar  to  that  which  occurs  in  the  continuous- 
current  transformer. 

At  certain  moments  during  the  revolution  of  the 
armature,  the  current  will  pass  direct  from  slip  ring  to 
commutator  brush,  and  at  all  other  times  the  transfer  of 
energy  may  be  considered  as  due  either  to  direct  conduc- 
tion through  the  windings  between  slip  rings  and  brushes, 
or  to  balanced  generator  and  motor  action — certain  por- 
tions of  the  winding  acting  as  generator  while  the  other 
portions  act  as  motor.  There  is  an  exception  in  the  case 
of  the  single-phase  rotary  converter,  the  action  of  which 
is  slightly  more  complicated ;  here  some  of  the  energy 
(about  one-third  of  the  total)  must  be  considered  as  trans- 
ferred by  unbalanced  generator  and  motor  action,  which 
involves  the  idea  of  a  flywheel  effect,  and  a  certain 
storage  of  energy  in  the  revolving  armature  during  por- 
tions of  a  complete  period,  this  energy  being  given  back 
again  in  the  form  of  direct  currents  at  other  instants. 
This  leads,  in  some  cases,  to  sparking,  and  it  is  one  of 
the  reasons  why  the  single-phase  rotary  converter  is  not 
so  satisfactory  a  machine  as  the  polyphase  converter. 

96.  Output  and  Efficiency  of  Rotary  Converters. 

—Without  entering  fully  into  the  question  of  I2  R  losses 
in  the  armature  windings  of  rotary  converters,  it  should 
be  understood  that  the  output  of  these  machines — except 


2/2  ASYNCHRONOUS   GENERATORS,    ETC. 

in  the  case  of  the  single-phase  converter — is  greater  than  that 
of  the  same  machine  used  merely  as  a  direct-current 
generator  ;  that  is  to  say,  the  I2  R  losses  are  appreciably 
less  than  if  the  whole  of  the  output  were  passed  through 
the  windings  in  the-  form  of  a  direct  current.  That  this 
must  be  the  case  is  fairly  obvious,  especially  when  it  is 
remembered  that,  as  the  connections  from  the  slip  rings 
pass  under  the  brushes,  there  is  a  direct  transfer  of  current 
from  slip  ring  to  brush  which  does  not  pass  through  the 
armature  conductors. 

The  output  of  a  rotary  converter  will  also  be  increased 
with  the  number  of  phases  or  slip  rings.  Thus,  the  I2  R 
losses  in  a  machine  provided  with  three  slip  rings,  and 
fed  by  a  three-phase  current,  will  be  greater  than  if  the 
same  machine  is  provided  with  six  slip  rings  and  fed  by 
a  six-phase  current.  This,  indeed,  is  what  would  be 
expected,  seeing  that  the  increased  number  of  phases 
reduces  the  current  per  phase  required  to  balance  the  con- 
tinuous current.  For  instance,  if  we  assume  a  constant 
direct-current  output  of  100  kw.  at  100  volts  and  a  power 
factor  of  unity  on  the  polyphase  side,  the  current  per 
phase  in  each  armature  section  for  a  total  input  of  100  kw. 
(all  losses  neglected)  would  be  544  amperes  for  a  three- 
phase  supply,  and  only  472  amperes  for  a  six-phase 
supply.  The  table  on  p.  273,  based  on  particulars  given 
in  Dr.  A.  S.  McAllister's  book  on  "  Alternating-Current 
Motors,"  gives  the  approximate  output  of  a  rotary  con- 
verter with  different  numbers  of  slip  rings ;  the  output  of 
the  same  machine  rated  as  a  direct-current  generator 
being  taken  as  unity.  The  power  factor  is  assumed  to  be 
100  per  cent. 

If  the  power  factor  is  less  than  unity,  the  output  will 
be  less ;  thus,  if  cos  6  =  '8  the  output  of  the  single-phase 
converter  would  be  only  7,  while  that  of  the  three-phase 


OUTPUT  OF  ROTARY  CONVERTERS      273 

and  four-phase  machines  would  be  about  1*15  and  1*4 
respectively. 

Another  advantage  resulting  from  an  increase  of  the 
number  of  phases  beyond  three  is  that  the  heating  of 
the  armature  conductors  is  more  evenly  distributed.  The 
PR  losses  are  not  evenly  distributed  throughout  the  arma- 
ture windings  of  a  rotary  converter  ;  but  with  an  increase 
in  the  number  of  phases,  there  will  not  only  be  less  local 
heating,  but  the  I2  R  losses  will  be  dissipated  more  uni- 
formly throughout  the  entire  armature. 

Since  the  output  of  a  rotary  converter  increases  with 


Number  of  Slip  Rings  or 
Phases. 

Output  of  Rotary 
Converter. 

Two  (single  phase)    ... 
Three  
Four  (two  or  four  phase)     ... 
Five      
Six 

•85 
I'35 
I-65 
I  -80 
1*0*5 

Eight    . 

2'O^ 

Ten 

2'!^ 

Infinity 

2-30 

the  number  of  slip  rings  or  phases  on  the  alternating- 
current  side,  it  is  obvious  that,  if  an  economical  and 
simple  means  can  be  employed  for  transforming  a  two 
or  three  phase  supply  into  a  polyphase  system  having  a 
larger  number  of  phases,  this  may  be  advantageous  under 
certain  conditions.  The  problem,  indeed,  is  not  a  difficult 
one,  as  many  solutions  can  be  obtained  by  a  suitable 
combination  of  vectors  differing  in  phase.  Fig.  98  illus- 
trates one  method,  known  as  the  "double  delta,"  by 
means  of  which  a  three-phase  supply  can  readily  be 
transformed  into  a  six-phase  system  suitable  for  feeding 
a  rotary  converter  provided  with  six  slip  rings.  The 

18 


274 


ASYNCHRONOUS   GENERATORS,    ETC. 


upper  diagram  shows  the  primaries  of  the  three  step- 
down  transformers  mesh-connected  between  the  supply 
mains.  Each  transformer  is  wound  with  two  equal  but 
distinct  secondaries,  and  the  six  sections  of  secondary 
winding  thus  obtained  are  connected  up  in  such  a 
manner  as  to  produce  a  six-phase  supply.  The  combi- 


nation of  vectors  is  shown  in  the  lower  diagram,  and  it 
will  be  seen  that  the  desired  result  is  obtained  by  mesh- 
connecting  the  coils  i,  3,  and  5  in  one  direction,  and  the 
coils  2,  6,  and  4  in  the  opposite  direction.  By  feeding 
the  slip  rings  from  the  junctions  A,  B,  C,  D,  E,  and  F, 
the  six-phase  supply  to  the  armature  is  obtained. 


OUTPUT  OF  ROTARY  CONVERTERS      2/5 

Messrs.  C.  P.  Steinmetz  and  L.  R.  Emmet  have 
devised  an  equivalent  arrangement  of  static  transformers, 
very  much  on  the  lines  of  Mr.  Scott's  phase-transforming 
system,  by  means  of  which  the  same  end  is  attained. 
This  is  known  as  the  "  double  tee  "  method. 

Mr.  A.  D.  Lunt  has  shown  how  a  twelve-phase  system 
can  be  obtained  in  a  similar  manner  from  a  three-phase 
supply,  but  the  supposed  advantages  of  increasing  the 
number  of  phases  to  this  extent  in  a  synchronous  con- 
verter are  not  obvious.  Even  in  the  case  of  very  large 
rotaries,  the  disadvantages  of  having  twelve  slip  rings, 
with  all  the  increased  complication  and  expense,  would 
seem  to  outweigh  the  possible  advantage  of  slightly  im- 
proved efficiency  or  increased  output.  The  theoretical 
output  of  the  twelve-phase  rotary  is  only  about  12  per 
cent,  greater  than  that  of  a  six-phase  machine. 

The  efficiency  of  rotary  converters  compares  favour- 
ably with  that  of  motor-generator  sets:  it  would  be 
about  90  per  cent.,  at  full  load,  for  a  three-phase  2oo-kw. 
machine  at  a  periodicity  of  40,  or  for  a  4oo-kw.  machine 
at  a  periodicity  of  60,  while  the  efficiency  of  an  equiva- 
lent motor-generator  set  would  not  be  much  above  86  per 
cent.  It  should  be  understood  that  these  figures  include 
the  losses  in  the  step-down  transformers  required  for  use 
with  the  rotary  converter  ;  but  the  motor  of  the  motor- 
generator  set  would  be  wound  for  the  high-tension  supply, 
which  might  be  of  the  order  of  6,000  volts. 

The  higher  frequency  machines  are  not  so  satisfactory 
as  those  built  for  the  lower  frequencies.  It  is  well  to 
remember  that  the  synchronous  converter  is  really  a 
compromise  between  a  direct -current  machine  and  a 
polyphase  generator  or  alternator,  as  the  case  may  be. 
There  is  no  objection  to  building  an  alternating-current 
generator  with  a  sufficient  number  of  poles  to  give  a 


2/6          ASYNCHRONOUS   GENERATORS,    ETC. 

frequency  of  60  cycles  ;  but  the  most  satisfactory  design 
of  large  direct  coupled  D.C.  generator  would  have  fewer 
poles  for  the  same  output,  the  frequency  of  currents  in 
armature  coils  being  of  the  order  of  8  to  12  ;  while  in 
the  case  of  high-speed,  belt -driven  machines  the 
frequency  may  well  be  as  high  as  20.  If,  therefore,  the 
rotary  converter  is  designed  for  a  frequency  of  25  (which 
is  usual  in  America),  the  number  of  poles  is  somewhat  in 
excess  of  what  would  be  considered  good  practice  in  a 
continuous-current  generator,  and  the  synchronous  con- 
verters of  50  and  60  cycles  are  necessarily  more  difficult 
to  design,  and  usually  of  less  satisfactory  performance. 

Tests  have  been  made,  and  figures  published,  tending 
to  show  that,  as  regards  the  all-day  efficiency  of  rotaries 
and  motor-generators,  there  is  not  much  to  choose  be- 
tween the  two;  but  the  cost  of  the  motor-generator  is 
generally  somewhat  higher  than  that  of  the  equivalent 
rotary  converter,  although,  when  the  necessary  step- 
down  transformers  and  regulators  are  taken  into  account, 
the  difference  in  this  respect  is  not  so  great  as  it  is  some- 
times supposed  to  be. 

97.  Starting  and  Synchronising  Rotary  Con- 
verters.— When  possible,  it  is  desirable  to  start  up  a 
rotary  converter  from  the  direct-current  side:  this  is  a 
simple  matter,  and  the  operation  is  in  all  respects  similar 
to  the  method  adopted  for  starting  up  an  ordinary  shunt- 
wound  direct-current  motor.  The  machine  is  then  syn- 
chronised by  varying  the  field  excitation  until  the  correct 
speed — as  indicated  by  the  synchroniser — is  obtained, 
when  the  switches  on  the  polyphase  side  can  be  closed. 
If  the  supply  is  three-phase,  it  is  usual  to  have  one  of  the 
switches  closed,  and,  at  the  moment  when  synchronism 
is  attained,  the  other  two  switches  are  thrown  in.  The 
operation  of  synchronising  has  to  be  carefully  carried 


STARTING   ROTARY   CONVERTERS  277 

out,  or  trouble  may  ensue.  In  cases  of  emergency,  the 
process  of  synchronising  may  be  omitted  ;  the  machine 
should  be  run  up  to  a  speed  slightly  above  synchronism, 
when  the  direct-current  switches  must  be  opened,  and  the 
switches  on  the  polyphase  side  immediately  closed,  thus 
connecting  the  armature  to  the  polyphase  supply.  The 
machine  will  pull  itself  into  step,  and  the  direct-current 
switches  can  then  again  be  closed. 

When  it  is  not  possible  to  start  up  from  the  direct- 
current  side,  a  very  satisfactory  method  consists  in  pro- 
viding a  small  induction  motor  for  the  purpose  of 
running  the  armature  up  to  speed.  This  auxiliary  motor 
is  designed  to  run  at  a  slightly  higher  speed  than  the 
rotary,  and  the  latter  is  then  brought  down  to  syn- 
chronous speed  by  loading  one  of  the  phases  through  a 
suitable  rheostat.  This  method  makes  the  process  of 
synchronising  exceedingly  simple,  and,  as  the  power 
developed  by  the  induction  motor  need  only  be  about 
one-tenth  of  the  full  load  output  of  the  rotary,  the  extra 
cost  is  not  very  great. 

There  is  another  method  of  starting  up  from  the  poly- 
phase side  which,  however,  is  only  applicable  to  small 
machines,  and  this  consists  in  switching  the  polyphase 
supply  directly  on  to  the  machine,  having  previously 
taken  the  precaution  to  disconnect  the  field  windings 
and  divide  them  into  several  sections :  the  main  switches 
on  the  direct  current  side  must,  of  course,  also  be  open. 
The  revolving  field  due  to  the  polyphase  currents  in  the 
armature  causes  the  machine  to  start  up  much  in  the 
same  way  as  an  induction  motor ;  and  when  synchronism 
is  reached — as  indicated  by  the  steady  reading  of  a  volt- 
meter across  the  direct-current  brushes — the  field  circuit 
may  be  closed ;  but  certain  precautions  must  be  taken  to 
ensure  that  the  field  circuit  is  closed  on  the  phase  which 


2/8  ASYNCHRONOUS   GENERATORS,   ETC. 

will  give  the  right  polarity  across  the  direct-current 
brushes.  The  object  of  dividing  up  the  field  winding 
into  several  distinct  sections  is  to  prevent  the  accumu- 
lated E.M.F.  induced  in  these  windings  during  the 
period  of  running  up,  reaching  such  a  high  value  as  to 
risk  a  breakdown  of  insulation.  One  objection  to  this 
method  of  starting  is  that  a  very  large  current  is  taken 
from  the  supply  mains,  amounting  to,  perhaps,  three 
times  the  normal  full-load  current  of  the  rotary,  and 
since  this  current  is  very  much  out  of  phase  with  the 
supply  pressure,  the  regulation  of  the  polyphase  system 
will  be  seriously  affected.  It  is  true  that,  before  switch- 
ing on  the  supply,  provision  can  be  made  for  reducing 
the  voltage  across  slip  rings  by  taking  intermediate 
tappings  from  the  low-tension  side  of  the  step-down 
transformers,  but  the  extra  cables  and  switch  gear 
required  are  a  somewhat  serious  objection. 

98.  Regulation  of  Rotary  Converters.— Since  the 
direct-current  pressure  across  brushes  has  a  definite  value 
for  a  given  alternating  pressure  across  slip  rings,  it 
follows  that  the  regulation  cannot  be  effected,  as  in  the 
case  of  a  motor-generator,  by  merely  altering  the  strength 
of  the  field  current.  Any  alteration  in  the  current  pass- 
ing through  the  field  windings  is  immediately  met  by  an 
alteration  in  the  armature  current,  which  so  adjusts 
itself  as  to  leave  the  resultant  field  as  before — i.e.,  of 
such  a  value  as  to  produce  the  required  back  E.M.F. 
corresponding  to  the  speed  of  synchronism. 

If  the  pressure  of  the  polyphase  supply  is  raised  or 
lowered,  then  the  direct-current  pressure  will  be  varied 
in  a  corresponding  manner.  This  suggests  what  is, 
perhaps,  the  most  perfect  method  of  regulation  :  namely, 
the  employment  of  variable  ratio  step-down  transformers, 
by  means  of  which — even  with  constant  pressure  at  the 


REGULATION  OF  ROTARY  CONVERTERS   2.79 

high-tension  terminals  of  the  polyphase  supply — the  volts 
across  slip  rings  can  be  varied  within  the  required  limits. 
This  regulation  can  be  done  by  hand  or  automatically ; 
but  automatic  regulation  on  these  lines  would  be  costly 
and  liable  to  give  trouble  owing  to  complication  of  parts. 

Another  method  of  varying  the  pressure  across  slip 
rings,  which  is  automatic  in  its  action,  is  of  considerable 
interest;  and  since  it  is  largely  used  in  practice,  it  is 
advisable  that  the  principles  involved  be  clearly  under- 
stood. In  the  first  place,  this  method  requires  the  com- 
pound winding  of  the  field  magnets  ;  but,  from  what  has 
already  been  stated,  it  is  evident  that  the  mere  strength- 
ening of  the  field  ampere  turns  as  the  load  increases  will 
not  produce  the  required  pressure  rise  across  slip  rings  to 
compensate  for  the  volts  lost  in  overcoming  the  resistance 
of  the  windings,  except  under  special  conditions  which  permit 
of  this  result  being  attained.  These  conditions  are  that 
there  must  be  an  appreciable  amount  of  self-induction  in 
the  circuit  between  the  high-tension  supply  terminals  (at 
constant  pressure)  and  the  slip  rings  of  the  rotary,  and 
that  the  ohmic  resistance  of  this  portion  of  the  circuit 
must  be  relatively  small.  If  the  step-down  transformers 
are  of  a  type  having  a  fair  amount  of  magnetic  leakage, 
it  may  not  be  necessary  to  provide  additional  self-induc- 
tion in  the  form  of  choking  coils ;  but,  in  practice,  small 
choking  coils  in  each  of  the  polyphase  leads  are  usually 
inserted. 

In  Fig.  99,  the  full  line  curve  shows  the  relation  be- 
tween armature  current  and  field  excitation  for  a  rotary 
converter  or  synchronous  motor  running  light.  The 
ordinates,  or  vertical  measurements,  indicate  the  amount 
of  the  armature  current,  while  the  horizontal  measure- 
ments represent  the  ampere  turns  on  the  field  magnets. 
There  is  a  certain  value  of  the  field  current — denoted  on 


280  ASYNCHRONOUS  GENERATORS,   ETC. 

the  diagram  by  the  letter  S — for  which  the  armature 
current  is  a  minimum ;  the  power  factor  is  then  approxi- 
mately unity,  and  the  very  small  armature  current  pass- 
ing into  the  windings  is  merely  what  is  necessary  to  run 
the  machine  light.  If,  now,  we  imagine  the  field  current 


Field  Ampere  Turns 

FIG.  99. 

to  be  reduced  to  a  value  indicated  by  the  distance  O  M 
in  Fig.  99,  the  armature  current  will  immediately  rise  to 
the  corresponding  value  M  N,  and  this  will  be  a  lagging 
current,  the  principal  component  of  which  is  a  magnet- 
ising current  90  degrees  behind  the  impressed  E.M.F., 


REGULATION  OF  ROTARY  CONVERTERS   28 1 

because  this  will  produce  a  flux  in  the  same  direction  as 
the  flux  due  to  the  direct  current  in  the  magnet  coils,  and 
its  value  will  be  such  as  to  provide  the  magnetising 
ampere  turns  which  have  been  taken  off  the  field  coils. 
If,  on  the  other  hand,  we  strengthen  the  field  current, 
the  main  component  of  the  armature  current  will  lead  the 
impressed  E.M.F.  by  90  degrees,  and  so  counteract  the 
magnetising  force  of  the  additional  field  ampere  turns. 
For  a  given  value,  O  P,  of  the  field  current,  the  no-load 


N 


FIG.  100. 


armature  current  may  be  of  exactly  the  same  value  as 
when  the  field  current  was  O  M  ;  but  in  the  first  case  it 
will  be  in  advance  of  the  impressed  E.M.F.  by  nearly 
a  quarter  period ;  while  in  the  latter  case  it  will  lag 
behind  the  impressed  E.M.F.  by  the  same  phase  angle. 

Now  consider  the  vector  diagram  Fig.  100.  Here 
O  E  represents  one  phase  of  the  impressed  polyphase 
E.M.F.,  which  is  supposed  to  be  of  constant  value. 
We  shall  not  complicate  the  diagram  by  taking  into 
account  the  step-down  transformers,  and  the  length  O  E 


282  ASYNCHRONOUS   GENERATORS,   ETC. 

must  be  considered  as  representing  the  primary  pressure 
reduced  to  the  pressure  at  slip  rings,  in  accordance  with  the 
transformer  ratios,  for  the  condition  of  the  correct  field 
excitation — i.e.,  when  the  field  current  is  such  as  to  make 
the  power  factor  a  maximum  (O  S  in  Fig.  99).  Under 
this  condition  the  armature  current  is  practically  zero 
Let  us  now  see  what  will  be  the  pressure  at  slip  rings 
with  a  reduced  field  excitation  and,  therefore,  a  lagging 
armature  current,  which  may  be  represented  by  the 
vector  O  N.* 

Note,  first,  that,  if  there  were  no  self-induction  in  the  circuit, 
the  pressure  across  slip  rings  would  remain  unaltered 
(except  for  the  small  resistance  drop,  which  may  be 
neglected);  but  if  the  circuit  has  self-induction,  the 
back  E.M.F.  due  to  the  current  O  N  will  be  O  e', 
drawn  exactly  90  degrees  behind  O  N.  The  result  will 
be  that  the  pressure  at  slip  rings  will  be  reduced  to  O  E', 
which  is  less  than  O  E  by  the  amount  O  e' .  Suppose 
now  that  the  field  current  is  increased  until  the  armature 
current  is  a  leading  one  of  value  O  R  ;  then  O  e"  will  be 
the  induced  E.M.F.  due  to  this  current,  and  it  will  be  in 
phase  with  O  E  :  the  pressure  at  slip  rings  will,  therefore, 
be  O  E",  which  is  equal  to  O  E  +  O  e". 

We  are  now  in  a  position   to   plot  the  dotted  curve 

*  In  this  diagram  (Fig.  100)  the  current  vectors  are  drawn 
exactly  90  degrees  out  of  phase  with  the  pressure  vector,  although 
in  practice  this  condition  is  never  quite  fulfilled.  But,  with  the 
machine  running  light,  the  phase  angle  will  be  very  nearly 
90  degrees,  and.  in  any  case,  even  if  the  machine  were  loaded,  it 
would  be  correct,  for  the  purpose  of  this  argument,  to  consider 
O  N  as  the  magnetising  component  of  the  total  armature  current, 
because  the  "  active  "  component,  in  phase  with  O  E,  will  produce 
only  cross  magnetising  effects,  and,  even  in  the  choking  coils,  will 
not  give  rise  to  a  back  E.M.F.  having  any  appreciable  effect  on 
the  regulation. 


REGULATION  OF  ROTARY  CONVERTERS   283 

in  Fig  99,  showing  how  the  pressure  at  slip  rings  in- 
creases with  increased  field  ampere  turns,  and  from  this 
it  is  easy  to  understand  that  a  rotary  converter  provided 
with  a  shunt  winding  across  the  brushes,  and  a  series 
winding  carrying  the  load  current,  may — by  suitably 
adjusting  the  amount  of  self-induction — be  made  prac- 
tically self-regulating  at  all  loads. 

It  is  important  to  bear  in  mind  that,  in  order  to  obtain 
this  result,  the  ohmic  resistance  of  the  cables,  trans- 
former windings,  etc.,  must  be  low ;  and  this  is  of  still 
greater  importance  if  it  is  desired  to  over-compound  the 
rotaries.  If  the  distance  of  transmission  is  great,  or  the 
voltage  so  low  as  to  involve  a  considerable  drop  of 
pressure  in  the  cables,  then  this  method  of  regulation 
is  unsuitable. 

The  chief  objection  to  the  compound  winding  of  rotary 
converters  as  above  described  is  that  the  power  factor 
of  the  polyphase  supply  is  continually  changing,  and  this 
makes  it  difficult  to  maintain  constant  pressure  at  the 
receiving  ends  of  the  transmission  lines.  Mr.  A.  C. 
Eborall,  in  his  paper  on  polyphase  substation  machi- 
nery,* states  that,  in  practice,  the  self-induction  in  the 
polyphase  leads  and  the  amount  of  the  field  excitation 
would  be  so  adjusted  as  to  make  the  armature  current  at 
no  load  a  lagging  one  equal  to  about  30  or  40  per  cent,  of 
the  full-load  current.  As  the  load  comes  on  the  machine, 
the  power  factor  improves  and  becomes  nearly  unity  at 
half  full  load,  after  which,  with  a  further  increase  of  load 
(and,  therefore,  of  field  ampere  turns),  the  armature 
current  would  be  in  advance  of  the  applied  E.M.F. 

99.  "  Hunting  "  of  Rotary  Converters.— The  satis- 
factory running  of  synchronous  motors,  and,  therefore, 

*  "Some  Notes  on  Polyphase  Substation  Machinery,"  Journ. 
Inst.  E.  E.,  vol.  xxx.,  p.  702. 


284  ASYNCHRONOUS   GENERATORS,   ETC. 

also  of  rotary  converters,  depends  largely  upon  the 
engines  in  the  generating  station.  If  these  have  not 
sufficient  flywheel  effect,  or  are  otherwise  defective  in 
the  matter  of  imparting  a  uniform  angular  velocity  to 
the  generators,  then  trouble  is  to  be  looked  for  in  con- 
nection with  synchronous  machines  taking  current  from 
the  supply.  By  uniformity  of  angular  velocity  is  meant 
constant  speed  per  revolution  ;  and  a  high-speed  engine 
will  necessarily  be  more  satisfactory  in  this  respect  than 
a  low-speed  engine  :  it  also  follows  that  "  phase 
swinging  "  or  "  hunting  "  troubles  are  not  likely  to  arise 
when  the  generators  are  driven  by  steam  or  water 
turbines. 

Sometimes  rotary  converters  will  "  hunt  "  even  when 
the  engines  are  in  all  respects  suitable  for  the  work  they 
have  to  do.  A  slight  oscillation  may  be  started  through 
various  causes,  and — especially  if  there  are  several 
machines  working  in  parallel — this  may  tend  to  increase, 
owing  to  periodic  distortion  of  the  field  produced  by  the 
"  swinging "  of  the  armature,  and  the  resulting  varia- 
tions in  the  phase  angle  of  the  current  fed  into  the  slip 
rings.  It  is,  therefore,  advisable  to  design  the  rotaries 
to  check,  as  far  as  possible,  this  tendency  to  "phase 
swinging  "  which  may  result  in  the  machines  falling  out 
of  synchronism,  not  to  mention  the  probability  of  severe 
sparking  at  the  commutator  brushes. 

Damping  coils  or  "  amortisseurs  "  suggest  themselves 
at  once  as  a  suitable  means  of  preventing  distortion  of 
the  magnetic  field  under  the  pole  faces  ;  and  these  coils 
may  take  the  form  of  extra  heavy  high-conductivity 
metal  flanges  supporting  the  field  coils  at  the  ends 
nearest  to  the  pole  faces.  The  use  of  such  castings, 
or  of  heavy  short-circuited  copper  conductors  between 
the  pole  shoes,  or  imbedded  in  the  pole  face,  may  lower 


"  HUNTING  "    OF   ROTARY   CONVERTERS        285 

the  efficiency  from  i  to  i|  per  cent.  ;  but  this  is  of  no 
consequence  compared  with  the  importance  of  steady 
running  and  freedom  from  hunting  troubles.  As  a 
matter  of  fact,  solid  cast-iron  pole  faces  act  in  a 
similar  manner  to  the  damping  coils,  and  are  equally 
effective. 

Hunting  is  far  less  likely  to  occur  with  low  than  with 
high  periodicities,  and  60  may  be  considered  as  the 
upper  limit,  beyond  which  it  would  be  inadvisable  to 
work  rotary  converters:  sparking  troubles  may  be  ex- 
pected, and — in  any  case  for  large  units —motor- gener- 
ators will  generally  be  found  preferable  at  this,  or  higher 
periodicities.  The  best  frequency  for  rotary  converters 
lies  between  25  and  40.  The  machines  work  well  on 
lower  frequencies,  but  the  cost  of  step-down  transformers 
becomes  excessive. 

Hunting  troubles  will  be  increased  if  the  connections 
on  the  polyphase  side  are  of  high  resistance,  and,  for  this 
reason,  rotary  converters  are  generally  found  to  work 
better  when  within  a  comparatively  short  distance  of  the 
generating  plant  than  when  placed  at  the  distant  end  of 
fairly  long  transmission  cables. 

In  conclusion,  it  should  be  stated  that,  although  the 
rotary  converter  has  been  considered  throughout  as 
taking  the  supply  from  the  polyphase  mains  (which 
is  the  usual  arrangement),  the  machine  is  reversible, 
and  can  be  supplied  with  direct  current  at  the  brushes, 
in  which  case  it  will  give  out  alternating  currents  at 
the  slip  rings.  If  such  a  machine  be  used  for  supplying 
alternating  or  polyphase  currents  in  parallel  with  other 
machines,  difficulties  arise  owing  to  the  absence  of 
armature  reaction.  Any  lagging  currents  on  the  alter- 
nating-current side  tend  to  weaken  the  field,  and  generally 
produce  a  condition  of  instability,  requiring  special 


286 


ASYNCHRONOUS   GENERATORS,   ETC. 


methods  of  field  excitation  to  remedy  the  trouble  and 
prevent  the  rotary  falling  out  of  step. 

100.  Motor  Converters. — The  La  Cour  Motor  Con- 
verter consists  of  two  machines  with  a  common  shaft. 
The  input  machine  (No.  i)  is  an  induction  motor  with 
three  slip  rings  for  starting  purposes,  and  a  wound  rotor 
which  feeds  current  into  the  armature  of  machine  No.  2. 
The  output  machine  is  a  synchronous  converter  provided 
with  a  commutator,  all  as  described  in  article  94.  The 


Stator 


Rotor 


Input  Machine. 

(  Induction  motor  operating  as  frequency 
converter  and  synchronous  motor.) 
Supplied  with  polyphase  currents. 


Output  Machine. 

(Combined  synchronous  converter 

and  D.C.  Generator.) 
Delivering  continuous  currents. 


FIG.  ioi. 


diagram  of  connections,  Fig.  ioi,  will  serve  to  explain 
the  principle  of  action  of  the  motor  converter.  This 
diagram  shows  the  rotor  of  the  input  machine  wound  for 
only  three  phases,  and  feeding  the  armature  of  the  out- 
put machine  through  connecting  wires  (carried  along 
or  inside  the  common  shaft)  terminating  at  points  on 
the  armature  120  electrical  space-degrees  apart.  The 
diagram  would  appear  to  indicate  that  the  output  unit 
(the  synchronous  converter)  is  a  two-pole  machine; 
but  it  can  obviously  be  designed  with  any  suitable 


MOTOR   CONVERTERS  287 

number  of  poles ;  and,  further,  since  the  synchronous 
converter  is  a  more  efficient  and  satisfactory  machine 
as  the  number  of  phases  of  the  supply  current  is  in- 
creased, it  is  usually  supplied  with  six  or  twelve  phase 
currents  in  the  practical  motor  converter.  The  increase 
in  the  number  of  phases  is  readily  accomplished  without 
appreciable  increase  in  cost,  since  it  is  merely  necessary 
to  take  the  requisite  number  of  taps  off  the  rotor  of 
No.  i  machine,  and  connect  these  to  the  corresponding 
points  on  the  armature  of  No.  2  machine.  It  may 
be  mentioned  that,  when  the  rotor  of  the  induction  motor 
is  wound  to  give,  say,  twelve-phase  currents,  it  is  not 
necessary  to  provide  more  than  three  slip  rings  for 
starting  purposes  :  the  motor  is  started  up  by  gradually 
cutting  out  the  external  resistance  from  the  three  phases 
only ;  but  all  twelve  windings  must  be  finally  short- 
circuited  when  full  speed  has  been  attained.  Bearing 
these  points  in  mind,  we  may  now  briefly  consider 
the  action  of  the  machine  on  the  basis  of  the  diagram 
Fig.  101. 

If  the  input  and  output  units  have  the  same  number 
of  poles,  the  speed  at  which  the  combined  set  will 
operate  will  be  exactly  half  the  speed  of  synchronism 
of  the  input  machine  considered  as  an  ordinary  induction 
motor  with  short-circuited  rotor  coils.  The  reason  of 
this  will  be  clear  when  it  is  considered  that  No.  2 
unit,  acting  as  a  synchronous  converter,  is  fed  with 
currents  having  a  frequency  proportional  to  the  slip 
of  the  rotor  of  No.  i  unit.  With  both  machines  at  rest, 
this  frequency  is  the  same  as  that  impressed  on  the 
stator  windings  of  machine  No.  i ;  but  as  the  speed 
of  rotation  increases,  this  frequency  decreases  until 
a  condition  of  balance  is  obtained,  when  the  speed  will 
remain  constant  for  a  given  impressed  primary  fre- 


288  ASYNCHRONOUS   GENERATORS,   ETC. 

quency.  The  exact  relations  between  speed  and  fre- 
quency and  relative  number  of  poles  are  best  explained 
with  the  aid  of  symbols. 

Let  f2  =  frequency  impressed  on  armature  of  machine 

No.  2; 
p2  =  number   of  pairs  of  field  poles   of  machine 

No.  2  ; 
R  =  actual  speed  of  revolution  of  both  machines 

(revolutions  per  second) ; 
/j  =  frequency  impressed   on   stator  windings  of 

machine  No.  i  ; 

pl  =  number  of  pairs  of  poles  on  No.  i  machine  ; 
Rj  =  synchronous  speed  (revolutions  per  second) 

of  rotor  of  No.    i   machine  considered  as 

induction  motor  with   secondary  windings 

short-circuited : 

then,  /2  =  R  x  pz 


slip  revolutions 
but/2=/lX-       -j^-      -«/! 

Substituting  this  value  of  /2  in  the  first  equation,  we 
get 

RR, 

A  -  (R!  - 

and  by  substituting  this  value  of  /t  in  the  second  equa- 
tion we  eliminate  the  frequencies,  and  get  the  relation 

R 


which  can  be  put  in  the  form 
R-     £ 


MOTOR  CONVERTERS  289 


or 


Thus,  the  combined  set  of  two  machines  has  a  definite 
speed  which  is  independent  of  the  load,  and  bears  a 
definite  relation  to  the  impressed  frequency  and  the 
number  of  poles  on  the  input  and  output  units.  When 
Pi  =  Pz  tne  sPee<i  is  obviously  exactly  half  that  of  the 
revolving  field  of  the  input  machine,  as  already  mentioned. 

The  action  of  the  two  machines  connected  together 
in  the  manner  above  described  may  be  summed  up 
as  follows:  When  continuous  currents  are  taken  from 
the  brushes  of  the  output  unit,  the  power  is  supplied  by 
the  input  unit  in  two  ways  —  (i)  by  transformer  action, 
the  input  machine  being  considered  as  a  frequency  con- 
verter ;  and  (2)  by  motor  action,  the  input  machine  being 
considered  as  a  synchronous  motor  transmitting  mechanical 
power  through  the  common  shaft.  The  output  machine, 
on  the  other  hand,  may  be  regarded  as  a  combination  of 
a  synchronous  converter  and  a  direct-current  generator. 
A  little  study  will  show  that  the  ratio  power  transmitted 
mechanically  to  power  transmitted  by  transformer  action  is  the 
same  as  the  ratio  p1  to  py 

The  chief  advantage  of  the  motor  converter  over  the 
synchronous  converter  lies  in  the  fact  that  the  corn- 
mutating  machine  can  be  supplied  with  polyphase 
currents  of  a  frequency  considerably  below  that  of  the 
primary  supply  circuit.  When  low  primary  frequencies 
are  used,  this  advantage  no  longer  exists  ;  but  the  motor 
converter  is  a  satisfactory  and  efficient  machine,  with  the 
starting  characteristics  of  the  ordinary  polyphase  motor. 
This  makes  the  starting  up  of  the  motor  converter  a  very 
simple  matter. 


APPENDIX  I 

ON    THE    RELATION    BETWEEN    MAGNETIC    FLUX   AND 
'  INDUCED    E.M.F.    IN    A    CIRCUIT    CONVEYING    AN 
ALTERNATING  CURRENT 

LET  N  be  the  total  amount  of  magnetic  flux  through 
a  coil  of  wire  due  to  a  certain  definite  maximum  value  of 
the  alternating  current.  We  need  not  concern  ourselves 
here  with  the  relation  between  the  current  and  the  total 
flux  ;  but  this  can  be  approximately  predetermined  in  the 
usual  way  (as  in  dynamo  and  motor  design),  provided 
the  material,  arrangement,  and  measurements  of  the 
magnetic  circuit  are  known. 

Let  S  be  the  number  of  turns  in  the  coil  of  wire, 
and  /  the  periodicity  of  the  current  (number  of  cycles  per 
second) ;  then,  since  the  total  number  of  magnetic  lines 
denoted  by  N  are  twice  created  and  twice  withdrawn 
during  the  course  of  one  complete  period,  the  mean  value 
of  the  induced  E.M.F.  will  be 

Ey=4NS/, 

and  this  equation  is  true  whatever  may  be  the  shape 
of  the  current  wave. 

In  the  above  formula  E°rt  is  the  induced  E.M.F. 
expressed  in  absolute  C.G.S.  units.  If  we  wish  to  ex- 
press the  mean  induced  E.M.F.,  Ea,  in  volts,  we  must 
write 

4NS/ 

*  IOO,OOO,OOO 

290 


APPENDIX   I  291 

It  is  not,  however,  the  mean  value  of  the  induced 
E.M.F.  which  we  generally  require  to  know,  but 
its  x/mean  square  value.  Let  us  denote  the  ratio 

V  mean  square  volts  ,  f 

-  -  —  -.  —  —  —  by  the  letter  m,  which  therefore 
mean  volts 

stands  for  the  quantity  usually  referred  to  as  the  form 
factor  :  we  may  now  write, 

Induced  E.M.F.  (in  volts)  =  E 


If  the  E.M.F.  wave  is  a  sine  curve,  the  form  factor  is 

2   7T/N    S 


Consider  now  a  circuit  in  which  the  flux  N  is  directly 
proportional  to  the  strength  of  the  current  I  which  pro- 
duces it,  as,  for  instance,  a  solenoid  or  coil  without  iron 
core.  The  constant  coefficient  of  self-induction  can  be  ex- 
pressed in  terms  of  N,  S,  and  I,  its  value  being 

_  flux  linkages 
current 

which,  expressed  in  henrys,  is 


where  lm  is  the  maximum  value  of  the  current  wave 
corresponding  to  the  production  of  the  maximum  number 
of  magnetic  lines  N. 

On  the  sine  wave  assumption,  lm  =  V^  I,  where  I  is 
the  R.M.S.  value  of  the  current ;  and  the  above  formula 
can  be  written 

N  S  =  10*  L  I 


292  APPENDIX   II 

If  this  value  for  the  total  flux  linkages  be  inserted  in  the 
voltage  equation,  we  get 

E    =    2    7T/L    I, 

which  is  the  well-known  formula  for  reactance  voltage  as 
used  in  connection  with  the  analytical  solution  of  alter- 
nating-current problems. 


APPENDIX  II 

METHOD  OF  DRAWING  THE  COMPLETE  VECTOR  DIAGRAM 
FOR  A  POLYPHASE  INDUCTION  MOTOR  FROM  THREE 
SETS  OF  MEASUREMENTS  MADE  ON  THE  MACHINE 

THE  three  sets  of  measurements  required,  in  order  that 
all  conditions  of  working  may  be  predetermined,  are  as 
follows : 

(1)  The  current,  volts,  and  true  watts  per  phase  of  the 
stator   winding   when    the   motor   is   running   light,   at 
practically  synchronous  speed. 

(2)  The  current,  volts,  and  true  power  per  phase  of  the 
stator  winding,  with  rotor  short-circuited   and   held  in 
fixed  position  to  prevent  turning. 

(3)  The  resistance  per  phase  of  the  stator  winding. 
The  set  of  readings  (i)  should  be  made  with  the  full 

working  voltage  on  primary  terminals ;  but  for  the  set  of 
readings  (2)  the  volts  at  primary  terminals  should  be 
reduced,  to  allow  of  a  current  not  exceeding  the  normal 
full-load  current  to  pass  :  in  both  cases  the  frequency 
must  be  the  same  as  that  for  which  the  motor  is  designed, 
and  on  which  it  will  have  to  work. 

The  general  diagram,    Fig.   85,   which   appeared   on 
p.  221,  has  been  reproduced  here,  together  with  the  new 


APPENDIX   II 


293 


diagram  Fig.  85A,  which  explains  the  construction  from 
the  above  data. 

Draw  O  A  (Fig.  85A)  in  any  direction,  to  represent  the 
magnetising  current  when  motor  is  running  light ;  and 
O  B,  to  represent  the  applied  potential  difference  at 
stator  terminals  ;  the  angle  between  these  two  vectors 


FIG.  85. 


being  a,  which  must  be  such  that  cos  a  =  the  power 
factor,  or  ratio  of  true  watts  to  apparent  watts. 

Now  draw  B  H  parallel  to  O  A,  and  of  such  a  length 
as  to  represent,  to  the  same  scale  as  O  B,  the  volts  lost 
in  overcoming  the  ohmic  resistance  of  the  stator  wind- 
ings :  it  will  be  equal  to  (O  A)  x  R,  where  R  stands 
for  the  resistance  per  phase  of  the  primary  winding, 
referred  to  under  test  (3).  Join  H  O  and  produce  to  K, 


294 


APPENDIX   II 


making  O  K  equal  to  O  H.  This  gives  us  the  vector 
O  K  for  the  E.M.F.  of  self-induction  (which,  in  the 
rotor,  is  balanced  by  the  E.M.F.  of  rotation  in  the  mag- 
netic field  ;  with  the  result  that  there  is  no  appreciable 
current  in  the  rotor  conductors).  It  is  this  pressure 
vector  which — in  Fig.  85 — is  supposed  to  be  of  constant 
value.  We  can,  therefore,  draw  the  dotted  circles,  of 
radius  O  K  or  O  H,  from  the  centre,  O,  and — under  all 
conditions  of  load — the  vector  representing  the  total 
induced  E.M.F.  per  phase  of  the  primary  circuit  must 


lie  on  the  arc  of  circle  described  through  K,  while  its 
balancing  E.M.F.  component  of  the  total  primary  pres- 
sure will  lie  on  the  arc  of  circle  described  through  H. 

We  will  now  turn  our  attention  to  the  second  set  of 
measurements — i.e.,  the  terminal  volts  and  the  true  watts 
corresponding  to  a  definite  primary  current  I-p  obtained 
when  the  rotor  is  short-circuited  and  clamped  in  position 
to  prevent  rotation. 

Note,  in  the  first  place,  that  the  magnetising  current 
(in  the  phase  O  A)  will  be  so  small  as  to  be  negligible, 


APPENDIX   II  295 

being  merely  such  as  will  induce  in  the  rotor  the  very 
low  volts,  Er,  required  to  overcome  the  resistance  of  the 
windings.  It  follows  that  the  total  primary  current,  lv 
is  almost  exactly  equal  and  opposite  to  the  rotor  current, 
I2,  and  these  current  vectors  must,  therefore,  be  drawn 
respectively  in  the  phases  O  H  and  O  K. 

Now  draw  O  D  in  advance  of  O  lv  to  represent  the 
measured  potential  difference  per  phase  at  primary 
terminals,  the  angle  ft  between  the  two  vectors  being 
such  that  cos  /3  =  the  ratio  of  (measured)  true  watts 
to  apparent  watts,  and  produce  D  O  to  M,  making 
O  M  =  O  D. 

The  vector  O  D  may  be  considered  as  being  made  up 
of  two  components  at  right  angles ;  one  of  these,  in 
phase  with  the  current,  and  equal  to  M  E3,  is  required 
to  overcome  the  resistance  of  stator  and  rotor  windings, 
while  the  other  (equal  but  opposite  to  O  E3)  is  required 
to  balance  the  E.M.F.  of  self-induction  due  to  the  leakage 
magnetism  which  does  not  enter  the  rotor.  In  this 
manner  we  obtain  the  length  of  the  vector  O  E3,  cor- 
responding to  a  given  secondary  current,  I2 ;  and  for 
any  other  value  of  the  rotor  current,  this  leakage  E.M.F. 
vector  may  be  assumed  to  vary  directly  as  the  current. 

Referring  again  to  the  vector  E3  M,  this  is  made 
up  of  two  parts,  M  N  and  E3  N,  representing  the 
E.M.F.  components  required  to  overcome  primary  and 
secondary  resistances  respectively.  It  is  easy  to  calcu- 
late M  N,  since  it  is  equal  to  (O  Ix)  x  R,  where 
R  =  primary  resistance  per  phase,*  and  this  leaves  E3  N, 

*  With  a  mesh-connected  winding  it  is  the  "  equivalent  "  star 
resistance  that  should  be  taken.  The  exact  meaning  of  this  quantity 
is  discussed  in  a  note  on  stator  winding  resistance  measurements, 
contributed  by  the  writer  to  the  Journal  of  Electricity,  of  San  Fran.' 
cisco,  May  9,  1914. 


296  APPENDIX    II 

or  O  E,.  for  the  E.M.F.  required  to  overcome  the 
secondary  or  rotor  resistance.  (It  is  interesting  to  note 
that  we  have,  in  this  way,  arrived  at  the  equivalent  re- 
sistance of  the  rotor  conductors  without  making  a  direct 
measurement  of  same.) 

In  order  to  complete  the  diagram  corresponding  to 
the  working  conditions  when  the  rotor  current  is  I2, 
produce  E3  M  (parallel  to  O  K)  until  it  meets  the  dotted 
circle  at  E" :  drop  the  perpendicular  E"  E2  on  to  O  K, 
and  produce  E"  O  to  EI}  making  O  Ej  equal  to  O  E". 

The  vector  O  E2  is  the  induced  E.M.F.  in  rotor 
under  working  conditions,  and  it  will  require  a  magne- 
tising current  O  lm  to  produce  it,  of  such  a  value  that 
O  \m  bears  the  same  relation  to  O  A  as  O  E2  bears 
to  O  K.  This  enables  us  to  obtain  the  primary  current 
vector  O  I.  Now  draw  Ex  E  parallel  to  O  I,  and  equal 
to  (O  I)  x  R,  where  R  =  primary  resistance.  Join 
O  E  :  then  O  I  and  O  E  are  the  vectors  representing 
the  primary  current  and  potential  difference  at  terminals 
which  will  be  required,  under  working  conditions,  when 
the  rotor  current  has  the  value  I2 ;  the  power  factor 
being  evidently  cos  0,  where  0  is  the  angle  subtended  by 
these  vectors. 

In  this  manner  the  complete  diagram  similar  to  Fig.  85 
can  be  constructed  from  the  three  sets  of  measurements 
referred  to  above ;  and  it  is  evidently  only  necessary  to 
carry  out  the  construction  for  one  more  value  of  the 
rotor  current  (equal  to  about  2  I2) — no  additional  tests 
being  required  for  this  purpose — in  order  to  obtain  three 
points  on  the  circle  diagram,  which  can  then  be  con- 
structed in  all  respects  as  shown  in  Fig.  89  on  p.  232. 


INDEX 


' '  ACTIVE  ' '  component  of  current 

or  E.M  F.,  118,  131 
Alternating  current,  i 

maximum,    mean,    and 

R.M.S.  value  of,  5 
Alternation,  5 
Alternator,    voltage    regulation 

of,  91 
Alternators,  parallel  running  of, 

96 

Aluminium    cell    lightning    ar- 
resters, 190 
conductors  for  transmission 

lines,  188 

Angle  of  lag,  12,  25,  46 
Apparent  power,  21,  25,  112 

resistance  (impedance),  46 
Armature  reaction,  91 

windings    of   polyphase 

generators,  82 
Asynchronous    generators,    91, 

247 
Auto- transformers    for    starting 

induction  motors,  237 
Average    value    of    alternating 
current  or  E.M.F. ,  5,  38,  126 

Capacity,  41,  55 
current,  57,  60 

of    transmission    lines, 

154,  158 
of  transmission   lines,   151, 

155,  156,  175 
of  underground  cables,  55, 

J45.  159 

Choking  coil,  125 
Circle    diagrams    for   induction 
motors,  226 


Circulating  currents  in  mesh- 
connected  armatures,  86 

Clock  diagrams,  10 

Coefficient  of  self-induction,  39, 
291 

Common  return  for  two-phase 
system,  90 

Comparison  between  different 
systems  of  transmission,  168, 

183 
Compensated  polyphase  motors. 

254.  257 
Compounding  synchronous 

generators,  93 
Condenser,  56 

current,  57,  61 
E.M.F.,  58 

Condensive  reactance,  62 
Connections  of  polyphase  arma- 
ture windings,  82 
of  transformers,  134 
Converters,  rotary.    See  Rotary 

converters 
Corona,  143 
Cosine  0,  25 
Coulomb,  59 

Current  circulating  in  mesh-con- 
nected armatures,  86 
flow  in  circuit  without  in- 
ductance, 41 
in  inductive  circuit,  42, 

291 

magnetizing,  50,  203,  208 
transformers,  122 
"wattless,"  23 


Delta  connection,  83,  134 
Disruptive  voltage,  143 


297 


298 


INDEX 


Distance  between  wires  in  trans- 
mission lines,  187 
Double-current  generators,  270 

Economical    section    of    trans- 
mission lines,  176 
Eddy  currents,  47 
Effective  value   of    E.M.F.    or 

current,  8 
Efficiency  of  polyphase  motors, 

222 
"Electromagnetic  momentum," 

29 

Electro-static  capacity,  55 
E.M.F.    induced    in    generator 

windings,  81 
in  motor  windings,  201, 

208 

in     transformer    wind- 
ings, 126 
of  self-induction,  31,  38,  45, 

290 
produced  by  an  alternating 

magnetic  field,  34 
E.M.F.s,    alternating,    addition 

of,  13,  17 

' '  Energy  ' '  component  of  cur- 
rent, 118,  126 

"Equivalent"  resistance  of 
stator  windings,  295 

Farad,  59 

Flywheel,  analogy  with  magnetic 

field,  29 

Form  factor,  60,  81,  291 
Frequency,  4 

converters,  255 

natural,  162 

of  generator,  79 

of    induction    motor    field, 
201,  218 

Generators,    asynchronous,   91, 

247 

compounding  of,  93 
regulation  of,  91 
synchronous,  78 

parallel      running     of, 
96 


Graphic  representation  of  alter- 
nating current,  2 
"Guard  wire,"  189 

Henry,  39 

Heyland  circle  diagrams,  226 
Heyland's   method  of  compen- 
sating induction  motors,  254 
Hysteresis,  32,  34,  48 

Impedance,  46,  63 

natural,  162 
Inductance,  38 

of  transmission  lines,   151, 

156 

Induction  motors.     See  Motors 
Inductive  load,  balanced  three- 
phase,  113 
effect  of,  on  line  losses, 

148,  169 

on  transformer,  132 
unbalanced      three- 
phase,  118 
Insulators    for   overhead    lines, 

189 
Iron  cores,  losses  in,  48,  50 

Kelvin's  law  of  economy,  177 

La  Cour  motor  converter,  286 

Leakage,  magnetic.  See  Mag- 
netic leakage 

Lenz's  law,  34 

Lightning,  protection  of  trans- 
mission lines  against,  189 

Lunt's  system  of  phase  trans- 
formation, 140 

Magnetic  field,  28,  31,  34,  92 
leakage  in  induction  motors, 

202,  218,  245 
in     transformers,    125, 

J95 

Magnetizing   current   of   choke 

coil,  50 
of  induction  motor,  203, 

204,  208 

of  transformer,  128 
Maxwell,  39 


INDEX 


299 


Mean     power     of     alternating 

current,  21 
value  of  alternating  current 

or  E.M.F.,  6,  38,  126 
Mesh  connection,  83,  108 
Motor  converters,  286 
Motors,    induction,    "compen- 
sated," 254,  257 
efficiency  of,  222 
method      of      drawing 
vector  diagram  from 
test  data,  292 
methods  of  starting,  234 
output  of,  212 
overload    capacity    of, 

213 
reversing    direction    of 

running,  244 
shape  of  slots,  245 
speed  regulation  of,  215, 

239 

theory  of,  74,  193,  198 
synchronous,  theory  of,  97 
Mutual  induction  in  transmission 
lines,  191 

Neutral  point,  artificial,  no 

of   star-connected    sys- 
tem, 87,  187 

Overhead     transmission     lines. 
See  Transmission  lines 

Parallel  running  of  alternators, 

96 

Periodicity,  4 
Periodic  time,  4 
Phase  difference,  8,  12 

transformation,  137 
Polygon  of  forces,  19 
Polyphase  generators,  78 
motors,  193 

output  of,  100 
regulation  of,  91 
transformers,  132 
Power,  apparent,  21,  25 
factor,  25,  46,  115 

of  polyphase  induction 
motors.  222,  230,  264 


Power    factor    of    three-phase 

circuit,  112 

in  three-phase  circuit,  106 
in  two-phase  circuit,  105 
mean,  in  alternating  current 

circuit,  21,  105 
measurement,  103,  122 
transmission,  142 

by    single  -  phase    cur- 
rents, 145,  156 
by  three-phase  currents, 

165 
by  two-phase  currents, 

163 
losses  in,  143,  166,  179, 

186 
Pressure-drop   on    transmission 

lines,  150,  163,  166 
-rise  on  transmission  lines, 
160 

Quantity  of  electricity,  58 

Reactance,  39,  62 
"  Reactive"  component  of  cur- 
rent, 118 

Regulator,  Tirrill,  94 
Residual  magnetism,  31,  34 
Resistance,  apparent  increase  of, 
with  alternating  currents,  146 
Resonance,  161 
Rotary  converters,  264 

efficiency  of,  275 
"  hunting  "  of,  283 
phase    transformation 

for,  273 
pressure  between  phases 

in,  268 
pressure  regulation  of, 

278 

starting  and  synchroniz- 
ing, 276 
theory  of,  269 

Rotating  magnetic  field,  65,  74 
Rotor,  76,  198 

Scott's  system  of  phase  trans- 
formation, 137 
Self-induction,  30,  39 


300 


INDEX 


Self-induction  of  single-phase 
transmission  lines, 
149 

of   three-phase   trans- 
mission lines,  172 
Series-parallel  control  of  motors, 

241 

transformers,  122 
Sine  waves,  10 

Single-phase  currents  from  poly- 
phase machines,  101 
transmission,  145,  156 
"Skin  effect,"  146 
"  Slip"  of  induction  motors,  76, 

214 

of     induction      motors, 

methods  of  measuring,  223 

Speed   regulation   of    induction 

motors,  215,  239 
Square  -  root  -  of  -  mean  -  square 
value  of  alternating  current,  5 
Star  connection,  87,  108,  134 
resistances,  calculation  of, 

no 

Stator,  76 

Synchronism,   speed  of  (induc- 
tion motors),  200 
speed  of  (motor  converter), 

287 

Synchronizers,  99 
Synchronous    generators.      See 

Generators 
polyphase  motor,  97 

Third  harmonic,  87,  89 

Three-phase  currents,  73 

delta  connection,  83 
star  connection,  87 
transformers,  134 

Time  angle,  12 
constant,  103 

Tirrill  regulator,  94 


Torque,  maximum,  213 

of  induction  motor,  200,  205, 

238 
Transformer,  123 

magnetic  leakage  in,  125 
theory  of,  125 
Transformers,  efficiency  of,  136 

polyphase,  132 
Transmission  lines,  143 

choice  of  voltage,  144 
comparison  of  different 

systems,  168,  183 
distance  between  wires, 

187 

economics,  168,  176 
height    above    ground, 

188 

pressu  e-rises  on,  160 
single-phase,  145,  etc. 
three-phase,  165 
two-phase,  163 
Triangle  of  forces,  18 
Two  -  phase    armature    connec- 
tions, 89 
currents,  66 
transformers,  133 

Underground  cables,  55,  61,  145, 
159,  161 

Vector  diagrams  explained,  14 
Vector  quantity,  definition,  14 
Virtual  value  of  E.M.F.  or 

current,  8 

Voltage,    choice    of,    on    trans- 
mission schemes,  144 
Volt-amperes,  26,  113 

11  Wattless  "  current,  23,  51,  118, 

126,  204 
Wattmeter,  103 
Wave  shapes,  non-sinusoidal,  60 


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